spc presentaion in word

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 UTOMOTIVE SECTOR CONTINUING PROCESS CONTROL & PROCESS CAPABILITY IMPROVEMENT A Guide to use of Control Chrts for !ulit" I#$ro%e#ent (Inten ded for use by Intern al Manufactu ring Facilit ies of Mahin dra & Mahin dra Ltd. and its Suppliers)  M!SSPCGL '( Re% '' )''*'( P+e ( of (*,  Printed copies of this document are not controlled

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CONTINUING PROCESS CONTROL

&

PROCESS CAPABILITY IMPROVEMENT

A Guide to use of Control Ch rts for !u lit" I#$ro%e#ent

(Intended for use by Internal Manufacturing Facilities of Mahindra & Mahindra Ltd.and its Suppliers)

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CONTINUING PROCESS CONTROL & PROCESS CAPABILITY IMPROVEMENT

Table of contents

Section Subject Page Number

I Introduction1. Preface2. Detection verses Prevention

3. Introduction to SPC4. nderstanding variation!. Data " #ariable and $ttribute%. Selecting Sam&le Si'e(. Probabilit) Distribution " t)&es*. Normal Distribution+. Control c,arts " tools for SPC

II Control c,arts1. -uidelines for selection of a&&ro&riate c,art2. #ariable Control c,arts /3. #ariable Control c,arts s4. #ariable Control c,arts 0/!. 0edian c,arts %. C,arts for individuals ( C l f $ ib P

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# $&&endi $ Ten e) ingredients to ma e SPC successful

in organi'ations.; <or ed e am&le =istogramC Coefficients for variables /D Coefficients for variables s> Coefficients for variables 0edian: Coefficients for variables Individuals- Critical values of t

= Critical values of χ2

I Critical values of :? #arious &robabilit) Distributions@ -lossar) of statistical termsA :ormulae of statistical terms0 5B6 $rea under normal curveN C,art format

$ttribute c,art format

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SECTION I . INTRO/UCTION

I0( PRE1ACENever ending im&rovement in ualit) Productivit) ,as become vital for survival intoda)6s business environment. In order to sustain ourselves in suc, economicclimate it is re7uired for us E 0a,indra 0a,indra Atd. and our Su&&liers todedicate ourselves to t,e tas of constantl) see ing more efficient a)s to &roduce&roducts and services t,at consistentl) meet customersF needs. To accom&lis, t,isGever)one in our organi'ations must bot, be committed to im&rovement and useeffective met,ods. T,is manual on 5C NTIN IN- P/ C>SS C NT/ A

P/ C>SS C$P$;IAITH I0P/ #>0>NT6 addresses some of t,e needs in t,isareaG b) describing basic statistical met,ods for im&roving our effectiveness.Different levels of understanding are needed to &erform different tas s t,is boo letis aimed at &ractitioners and managers beginning t,e a&&lication of statisticalmet,ods.T,e basic statistical met,ods addressed in t,is boo include t,ose associated it,statistical &rocess control and ca&abilit) anal)sis. T,e manual gives somebac ground of &rocess controlG e &lains several im&ortant conce&ts suc, ass&ecial and common causes of variationG and introduces t,e control c,artG a ver)effective tool for &rocess control. T,e manual also describes t,e construction anduse of control c,arts for variables data 87uantitative dataG or measurements9 J KJbar / c,artsG K 0/ c,artsG median c,arts. T,e manual includes severalcontrol c,arts for attribute data 87ualitative dataG or counts9 J t,e & c,artG n& c,artG cc,art and u c,art.T,e a&&endi includes table of constantsG glossar) of termsG s)mbolsG formulaeand re&roducible co&ies of blan control c,art forms.

l ) d f l

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I0- /E1ECT /ETECTION VERSUS /E1ECT PREVENTIOND>:>CT D>T>CTI N 8 after t,e event9J tolerates t,e aste

D>:>CT P/>#>NTI N 8before t,e event9J avoids t,e aste

Ti#el"

Not O6

Ins$e4tion8 AuditsChe49s & B l n4es

PEOPLEMATERIALSMET5O/SMAC5INESENVIRONMENT

TRANS1ORMATION PRO/UCTS ORSERVICES

O6

1eed: 49 ;/el "ed<

SCRAPor

RE7OR6

CUSTOMER

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/efe4t /ete4tionIt is a &ast oriented a&&roac, t,at attem&ts to identif) and se&arate unacce&table

out&ut from acce&table out&ut.

/efe4t /ete4tion.• Is reactionar)• Tolerates aste• /elies on ins&ectionG auditsG or c,ec s of large sam&les of out&ut• /eacts to all defects indiscriminatel)

• :ocuses on conformance to s&ecifications• Involves action onl) on out&ut• /elies on dela)ed feedbac for defect detection• Is not cost effective

/efe4t Pre%ention.It is a future oriented a&&roac, t,at im&roves 7ualit) and &roductivit) b) &recludingdefect generation/efe4t Pre%ention.• Is &roJactive• $voids aste• ses small sam&les of &roduct and &rocess information• Is anal)ticall) based• Discriminates bet een &otential defects based on causes• Involves action on t,e &rocess or &rocess &arameters

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I0) INTRO/UCTION TO STATISTICAL PROCESS CONTROL $ traditional a&&roac, to manufacturing is to de&end on &roduction to ma e t,e&roductG and on 7ualit) control to ins&ect t,e final &roduct and screen out items notmeeting s&ecifications. In administrative situationsG or is often c,ec ed andrec,ec ed in efforts to catc, errors. ;ot, cases involve a strateg) of detection. It is

astefulG because it allo s time and materials to be invested in &roducts or servicest,at are not al a)s usable. It is a &astJoriented a&&roac, t,at attem&ts to identif)and se&arate unacce&table out&ut from t,e acce&table out&ut. $fterJt,eJfactins&ection is uneconomical it is e &ensive and unreliable as it involves tolerance to

aste in &roducing defects J t,en finding t,e defectives E re&airing or scra&&ing t,edefectives.

<,ic, meansG t,e traditional met,ods of 7ualit) control in a manufacturing &rocess,ave been t,ose of 1 M ins&ection and sam&ling ins&ection. ;ot, met,odsconcentrate on t,e final out&ut of t,e &roduct.=o ever 1 M ins&ection is not 1 M reliable and on some occasions fullins&ection as to be carried out more t,an t ice. :or large consignments sam&lingins&ection as &referred. =o ever t,ere is al a)s a ris ofa9 ;atc, ma) be rejected alt,oug, it is not as bad as it a&&ears from t,e sam&le.b9 T,ere is a c,ance t,at a batc, is acce&ted alt,oug, it is orse t,an originall)

t,oug,t on t,e basis of t,e sam&le.T,is contradicts to t,e &rinci&le of total 7ualit) and t,ere ,as to be alternative tot,ese met,ods of 7ualit) control.

I i ff i id b) & d i bl & i fi

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• Peo&le• 0aterials• 0et,ods• 0ac,ines 8e7ui&ment9• >nvironmentT,e transformation

• T,e value added ste&• C,anges t,at are re7uired to convert in&ut into out&ut• ;lending of resourcesT,e &rocess out&ut

• Products• Services• Customer Satisfaction

$ll &rocesses generate data or information t,at can be used to im&rove t,e&erformance of t,e &rocess itself. It is asteful to use t,is information onl) to

,ig,lig,t t,e need for rectification on t,e nonJconforming &roducts. Informationabout t,e s)stem is essential to identif) im&rovement o&&ortunities so as tocontrol t,e stabilit) of t,e &rocess and t,us t,e &roduct consistenc).

$ &rocess control s)stem can be described as a feedbac s)stem. :our elementsof t,at s)stem are im&ortant to t,e discussions t,at ill follo L• The Pro4ess E Process means t,e ,ole combination of &eo&leG e7ui&mentG

in&ut materialsG met,odsG and environment t,at or toget,er to &roduce out&ut.T,e total &erformance of t,e &rocess J t,e 7ualit) of its out&ut and its &roductiveffi i ) J d & d ) & b d i d d b il G d

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• A4tion on the Out$ut J $ction on t,e out&ut is &astJorientedG because itinvolves detecting outJofJs&ecification out&ut alread) &roduced. nfortunatel)G ifcurrent out&ut does not consistentl) meet customer re7uirementsG it ma) benecessar) to sort all &roducts and to scra& or re or an) nonJconforming items.T,is must continue until t,e necessar) corrective action on t,e &rocess ,asbeen ta en and verifiedG or until t,e &roduct6s s&ecifications ,ave been c,anged.

Process Control is also based on t,e PlanJDoJC,ec J$ct C)cle E a model forContinuous Im&rovement. During Process ControlG t,e PJDJCJ$ c)cle is used toassess and manage #ariation.

P A $

N

D C C = > C @

$ C T

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To find out ,o t,e &o&ulation is be,avingG e ill observe onl) a &art of t,e&o&ulation and gat,er data. <e ill t,en use t,is data to infer somet,ing about t,e

&o&ulation. Sam&le is a subJset of t,e &o&ulation.:or e am&le1. T,e number of engineers it,in a de&artmentG in a com&an).2. T,e number of ,ouse,old in a bloc of a cit)G ,o ,ave com&uters.3. T,e cro& &roduction of a &articular variet) in a small village of t,e district.4. T,e number of devotees visiting a s,rine &er mont,.

Po$ul tion nd S #$le s $$lied to St tisti4 l Pro4ess Control0Po&ulation is a conce&t t,at means to im&l) and include all &ossible reali'ations of a&rocess it,in a certain frame. $ &o&ulation is reall) defined b) a constant s)stem ofvariation causes c,aracteri'ed b) a succinct set of &arameters t,at in an average orcollective sense describe t,e salient features of t,e variation s)stem.;) sam&ling e observe and collect data of onl) a &art of t,e &rocess. =o muc,

e sam&le eac, timeG ,o often andQor ,en e sam&leG and ,o e s&ecificall)ta e t,e sam&le are critical issues t,at must be addressed.#arious functions of t,e data ma) be used to calculate measuresG eac, of ,ic, is areflection of some s&ecial feature of &o&ulationQ&rocess. T,ese sam&le measuresare called statisticsG eac, of ,ic, ma) be used as an estimate of t,e corres&onding&o&ulation &arameter. Toget,erG a set of &arameter estimates is used toc,aracteri'e t,e &rocess &erformance.

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I02 UN/ERSTAN/ING VARIATION

No t?o thin+s re tot ll" li9e or identi4 l0 =ence variation is a natural&,enomenon and is universal.<e ,ave ,eard sa)ing to su&&ort t,is J fi%e fin+ers on our h nd re ll differentfro# one nother O. <e also sa) E @no t?o $ersons thin9 li9e .#ariation is t,e inevitable difference among individual out&uts of a &rocess. In orderto effectivel) use &rocess control measurement dataG it is im&ortant to understandt,e conce&t of variation.No t o &roducts or c,aracteristics are e actl) ali eG because an) &rocess containsman) sources of variabilit). T,e differences among &roducts ma) be largeG or t,e)ma) be almost e tremel) smallG but t,e) are al a)s &resent.T,e diameter of a mac,ined s,aftG for instanceG ould be affected b)G

M 4hine 8clearancesG bearing ear9Tool 8strengt,G rate of ear9M teri l 8c,emical com&ositionG ,ardness9O$er tor 8&art feedG accurac) of centering9M inten n4e 8lubricationG re&lacement of orn &arts9En%iron#ent 8tem&eratureG constanc) of &o er su&&l)9

:or anot,er e am&leG t,e time re7uired to &rocess an invoice could var) accordingto t,e &eo&le &erforming various ste&sG t,e reliabilit) of an) e7ui&ment t,e) ereusingG t,e accurac) and legibilit) of t,e invoice itselfG t,e &rocedures follo edG andt,e volume of ot,er or in t,e office.Some sources of variation in t,e &rocess cause ver) s,ortJrun &ieceJtoJ&iecedifferences J e.g.G bac las,G clearances it,in a mac,ine and its fi turesG or t,e

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values are different as a grou& t,e) tend to form a distribution &attern t,at can bedescribed as &redictable in terms of Aocation 8t)&ical value9G S&read 8amount b)

,ic, t,e smaller values differ from t,e larger ones9 and S,a&e 8t,e &attern ofvariation J ,et,er it is s)mmetricalG &ea edG etc.9.> am&les of Co##on C uses of variation includeG

#ariable ra materialsG/igid or ing met,odsG>7ui&ment limitationsG

$tmos&,eric conditionsGPersonnel ca&abilitiesG etc.

T,e e tent of Common Causes of variation can be indicated b) sim&le statisticaltec,ni7uesG but t,e causes t,emselves need more detailed anal)sis to isolate.T,ese common causes of variation are usuall) t,e res&onsibilit) of management tocorrectG alt,oug, ot,er &eo&le directl) connected it, t,e o&eration sometimes arein a better &osition to identif) t,ese causes and &ass t,em on to management forcorrection. verallG t,oug,G t,e resolution of common causes of variation usuall)re7uires actions on t,e s)stem.

Common Causes of #ariation• =ave fre7uenc) distributions t,at are stable over time• /esult in &redictable outcomes• $re &ermanent unless action is ta en on t,e s)stem• Can onl) be eliminated b) c,anging t,e Process or S)stem

Special Causes arise from causes t,at affect onl) some out&ut of t,e &rocess andl f ) f

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ntried Qne &roduction &rocessesC,anges in ins&ection met,ods

M teri ls0i ing of &artsG batc,es $ccumulation of aste &roducts or im&uritiesAac of ,omogeneit)C,anges of sources of su&&l)

En%iron#ent-radual deterioration in conditions.SeasonalG dail)G ee l) c,anges.#ariations in tem&erature ,umidit) ,ic, affects t,e &rocess#ariation of noise dust ,ic, affect t,e o&erator

S&ecial causes of variation can be detected b) sim&le statistical tec,ni7ues. T,esecauses of variation are not common to all t,e o&erations involved and are not a &artof t,e s)stem as designed. S&ecial causes of variation result in out&ut be,avior t,atis usuall) erratic and un&redictable. T,e discover) of a s&ecial cause of variationGand its removalG are usuall) t,e res&onsibilit) of someone ,o is directl) connected

it, t,e o&erationG alt,oug, management sometimes is in a better &osition tocorrect. T,e resolution of a s&ecial cause of variationG t,enG usuall) re7uires localaction. C,arting ,ig,lig,ts occurrence of s&ecial causes.S&ecial Causes of #ariation

• =ave fre7uenc) distributions t,at are unstable over time• /esult in un&redictable outcomes• /ea&&ear local unless action is ta en

f

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T,e follo ing c,arts include &ictorial re&resentations of a fe of t,e situations t,atcan occur and ,o t,e Process Performance can be inter&reted.

T I M E

T A R G E

T

VARIATION in #e n VARIATION in s$re d Pro4ess is OUT O1

CONTROL Pro4ess is NOT

STABLE COMMON as ell as

SPECIAL causes of VARIATION &resent

C

T I M E

T A R G E

T

VARIATION in #e n N VARIATION in s$re d N VARIATION in sh $e Pro4ess is NOT STABLE SPECIAL causes of

VARIATION $resentPro4ess is being i#$ro%ed

D

T I M E

T A R G E

T

NO variation in t,e #e n

NO variation in t,es$re d

NO variation in t,e sh $e T,e &rocess is STABLE nl) COMMON causes

of VARIATION

$

T I M E

T A R G E

T

O11 TARGET

NO variation in t,e #e nNO variation in t,e s$re d NO variation in t,e sh $e T,e &rocess is STABLE nl) COMMON causes of

VARIATION

;

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I03 /ATA

VARIABLE & ATTRIBUTEIn SPCG data serves as one of t,e im&ortant basis for corrective action. $s suc,G it isnecessar) during conducting SPC t,atG

• t,e sam&led data are re&resentative of t,e total &rocess out&ut 8sam&lingmet,ods9G and

• T,e data reflects facts so t,at ,en collectedG anal)'ed and com&aredG itreveals facts 8statistical &rocessing9.

Statistical tec,ni7ues deal it, observation anal)sis of sam&les and estimating t,econdition of t,e &o&ulation in ,ic, t,e sam&le is included. Sam&ling met,ods areem&lo)ed and are im&ortant to obtain t,e re7uired information from t,e sam&lesG sot,at it re&resents t,e &o&ulation. 0ost of t,e manufacturing &rocesses deal it,,eterogeneous dataG as suc, careful stratification is re7uired it, res&ect to timeGo&eratorG mac,iner)Qe7ui&mentG materialQlotsG or ing conditions suc, ass&eedsQfeedG etc.

>ver) &rocess generates data t,at can be categorised in 5#ariables data6 5$ttributes data6.V ri :le d t#ariables data relates to ,at can be measured and e &ressed 7uantitativel) ins&ecific units of measurements as dimensionsG volumeG tem&eratureG &ressureG timeGstrengt,G etc. T,e e am&les include

• diameter of a bearing• volume of t,e c)linder

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T,e conformance criteria must be clearl) defined and t,e &rocedureQs for deciding ift,ese criteria are met must &roduce consistent results over time. T,e follo ing table

of e am&les could be ,el&ful in t,ro ing some lig,t on t,is as&ectL

A44e$t n4e Criterion ;S$e4ifi4 tion< Co##entSurface s,ould be free from fla s • <,at is a fla R

• Do t,e ins&ectors understandR• Do t,e ins&ectors agreeR

• =o can it be measuredRSurface s,ould conform to 0asterStandard in colour te tureG brig,tnessand im&erfections

• Conform to ,at degreeR• =o can it be measuredR

$n) material a&&lied to mirror bac to&revent scattering s,all not causevisible staining of t,e mirror bac ing

• #isible to ,om Q b) ,atR• #isible under ,at conditionsR• Do t,e ins&ectors understandR• Do t,e ins&ectors agreeR

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I0* SELECTING SAMPLE SI E

5o? to sele4t s #$le siDe

ne of t,e first basic 7uestions ,ic, comes in our mind before carr)ing out asurve) or statistical stud) is =o man) sam&les s,ould I selectRO-enerall) e select t,e sam&les as &er our convenience or it,out an) &articularlogic e ce&t cost time.Since sam&ling is time consuming and costl)G our objective in selecting a sam&le isto obtain a s&ecified amount of information about a &o&ulation at a minimum cost.<e can accom&lis, t,is objective first deciding on a bound of error of estimation andt,en a&&l)ing an a&&ro&riate sam&le si'e estimation formula.If t,e &o&ulation is uniformG t,e small sam&le &rovides t,e same amount ofinformation as a large sam&le. :or e am&le a &,)sician can base a diagnosis onone or fe dro&s of &atient6s blood. Selecting a large sam&le in suc, case ould be

aste of mone) time. n t,e ot,er ,andG if t,e &o&ulation consists of man) ,ig,l)diverse elementsG a small sam&le ma) &rovide a &oor reflection of t,e &o&ulation.:or e am&leJ to stud) t,e estimate of t,e average ,eig,t of male students in acollege.Sam&le si'e decisions are made according to t,e in,erent variabilit) in t,e&o&ulation of measurements and ,o accurate t,e e &erimenter is,es t,eestimate to be. T,ese t o criteria are of course inversel) related. To obtain greateraccurac)G and ,ence more information about a &o&ulationG e must select a largersam&le si'e. t,e greater t,e in,erent variabilit) in t,e &o&ulationG t,e larger sam&leis re7uired to maintain a fi ed degree of accurac) in estimation.

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( )22 R n+e(*(=σ

-0 Str tified R ndo# S #$lin+<it, a stratified random sam&ling e select a se&arate sim&le random sam&le it,ineac, of t,e strata. T,erefore e cannot determine n until e no t,e relations,i&bet een n and t,e sam&le allocation to t,e strata n 1G n2 ..n A . <,en using stratifiedrandom sam&lingG e must also consider t,e fact t,at t,e variances of t,e strataG

0a) not be e7ual. <e ill need a&&ro imations for eac, oft,ese variancesG ,ic, e can obtain from &revious sam&les.

( )221 R n+e

(*(

Provides a roug, estimate of t,e variance of t,e measurements in stratum i basedon t,e range of measurements it,in stratum i.=ence

str tu#0thi

theof siDethend % ri n4ethe8l"res$e4ti%ereN nd 7here

2B / nd

N N(

N/

N n

i-i

-

L

(ii

L

ii

σ

σ

σ

=+

=

∑∑

=

=

2

1

2

i

i

0ost of t,e sam&le si'e calculations &resented assume a normal distribution. To

-L0000000008 σ σ σ

22

21 ,

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• Subgrou& s,ould ensure t,e &resence of normal distribution for t,e sam&lemeans

• Subgrou&s s,ould ensure good sensitivit) to t,e detection of s&ecial causes.• Subgrou&s s,ould be small enoug, to be economicall) a&&ealing from a

collection measurement stand&oint.

Generally 4 to 6 sample / subgroup is commonly usedconsidering the above factors.Note L• S,e ,art suggests 2 as t,e ideal subgrou& si'e• In industrial use of control c,art ! is most common• >ssential idea of control c,art is to select sub grou&s in suc, a a) t,at it gives

minimum o&&ortunit) for variation it,in sub grou&G it is desirable to be as smallas &ossible 8for economic &ur&ose9.

• $ si'e of 2 is better t,an 2 or 3 on statistical grounds t,e distribution of isnearl) normal for sub grou&s of four or more even t,oug, t,e sam&les are ta en

from non normal universe.• Sub grou&s of 2 or 3 are used NAH ,en cost of measurement is ,ig,

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I0= PROBABILITY /ISTRIBUTION

Data generated 8attributes variables9 ,en arranged gra&,icall) or in a tabularmanner for a &articular c,aracteristicG de&ending u&on t,e t)&e ould follo a&articular &atternG called as fre7uenc) distribution. Distribution can also becategorised asL

• Continuous Distribution J generall) evident it, variableFs data and• Discrete Distribution J associated it, t,e attributeFs data.

T,ere are various t)&es of Probabilit) Distributions. T,e) are as given belo1. Normal 8-aussian9 Distribution2. Aog Normal Distribution3. <eibull Distribution 8T o Parameter94. <eibull Distribution 8T,ree Parameter9!. > &onential Distribution%. ;inomial Distribution(. =)&ergeometric Distribution*. Poisson Distribution

(0 Nor# l ;G ussi n< /istri:ution• :ield of $&&licationL #arious P,)sicalG 0ec,anicalG >lectricalG C,emicalG etc.

c,aracteristics Q &ro&erties.• Test Cases L Ca&acit) variation of electrical condensersG tensile strengt, of

aluminum s,eetG mont,l) tem&erature variationG &enetration de&t, of steels&ecimensG s,aft diametersG electrical &o er consum&tion in a given areaG rodlengt,sG gas molecules velocitiesG noiseG resistanceG generator out&ut voltageG ind

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20 7ei:ull /istri:ution ;Three P r #eter<• :ield of $&&licationL Same as t oJ&arameter <eibull andG in additionG various

&,)sicalG mec,anicalG electricalG c,emicalG etc.G &ro&ertiesG e ce&t less commont,an in t,e case of Normal distribution.

• Test CasesL Same cases as <eibull 8t o &arameter9. In additionG electricalresistanceG ca&acitanceG fatigue strengt,G etc.

30 EF$onenti l /istri:ution• :ield of $&&licationL T,e life of s)stemsG assembliesG etc. :or com&onentsG

situations ,ere failures occur b) c,ance alone and do not de&end on timeJinJservice fre7uentl) a&&lied ,en t,e design is com&letel) debugged for &roductionerrors.

• Test CasesL Aife to failure of automotive transmissionsG e &ected cost to detectbad e7ui&ment during reliabilit) testingG e &ected life of indicator tubes in radare7ui&mentG e &ected life of electrical lig,ting tubes in a factor)G etc.

*0 Bino#i l /istri:ution

• :ield of $&&licationL Number of defectives in n sam&le si'e dra n from a lot,aving & fraction defectives &robabilit) of occurrences in a grou& of )occurrencesG i.e.G situations involving 5goJno go6G 5 @Jdefective6G 5goodJbad6 t)&es ofobservations. Pro&ortions of lot does not c,ange a&&reciabl) as a result of sam&ledra n.

• Test Cases L Ins&ection of defectives in a s,i&ment of steel &artsG ins&ection ofdefective com&onents in a &roduction lotG determination of defective eld jointsG&robabilit) of obtaining electrical &o er of a certain attage from a sourceG

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I0> NORMAL /ISTRIBUTION

<,ile t,e details of t,e distributions e ce&t Normal Distribution are not in t,e sco&eof t,is manualG onl) Normal Distribution s,all be discussed.T,is is also no n as -aussian DistributionG because of contribution of @arl -aussGeig,teent, centur) -erman mat,ematician.0an) c,aracteristics of manufacturing &rocesses ,ave a Normal Distribution.

Ch r 4teristi4s of Nor# l /istri:ution.

• It is s)mmetric distribution and is bell s,a&ed.• T,e mean defines ,ere t,e &ea of t,e curve occurs. In ot,er ordsG t,e

ordinate at t,e mean is t,e ,ig,est ordinate. T,e ,eig,t of t,e ordinate at adistance of one standard deviation from mean is % .%!3 M of t,e ,eig,t ofmean ordinate and similarl) t,e ,eig,t of ot,er ordinates at various standarddeviations from mean ,a&&ens to be a fi ed relations,i& it, t,e ,eig,t ofmean ordinate.

• T,e curve is as)m&otic to t,e base lineG ,ic, means t,at it continues toa&&roac, but never touc,es t,e ,ori'ontal line.• T,e variance defines t,e s&read of t,e curve.• T,e area enclosed bet een t o ordinates at one sigma 8SD9 from t,e mean

on eit,er side ould be al a)s %*.2*%M of total area.• T,e normal distribution ,as onl) one mode since t,e curve ,as onl) one

&ea . In ot,er ords it is unimodal distribution.• T,e ma imum ordinate divides t,e gra&, of normal curve into t o e7ual

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S5APE O1 NORMAL /ISTRIBUTION

NORMAL DISTRIBUTION

NORMALDISTRIBUTION

LOCATION(CENTRALITY) ( )

SPREAD (6 )

STANDARDDEVIATION ( )

X

X

X ~

A Normal Distribution has differentlocations :

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It is rat,er difficult to dra an) conclusions from t,e ra data b) visual e aminationsonl). Some means of summari'ing t,e information is re7uired. =ence a first ste& to

convert Qreduce mass ra data to a more com&re,ensive form is to define a numberof distinct and e ,austive categories into ,ic, observations ma) be divided. If ere&lace ra data b) listings of t,e grou&sG and number of &oints l)ing in eac,G calledfre7uenc). Suc, listing is called a fre7uenc) distribution. T,e im&ortant features of aset of data can be seen even more clearl) in a gra&,ic re&resentation of fre7uenc)distributionG called as =istogram.

$ Normal Curve is related to a fre7uenc) distribution and its =istogram. $ ,istogramsummari'es data from a &rocess and gra&,icall) &resents its fre7uenc) distribution

in a bar form.

GRAP5ICAL REPRESENTATION 5ISTOGRAM=istogram is t,e &rimar) assessment of &rocess centeringG s&read and s,a&e of t,edata.

uic ste&s in ma ing a ,istogramL• -at,er data 8realJtime observations9• Tabulate t,e data• Count t,e number of data &oints 5n6• Determine t,e range 5/6

• / Kma . 8ma imum reading9 E Kmin. 8minimum reading9• Determine t,e number of Class Intervals 5 6

• /Qn or • arbitraril)G 5 6 can be determined from t,e follo ing tableG

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Uses of histo+r #8• It &rovides information about distribution 8s,a&e9 of t,e data

• It &rovides information about s&read 8variation9 of t,e distribution• It &rovides information about location 8centralit)9 of t,e distributionN T>L It is im&ortant to no t,at a =istogram is not a ;ar C,art bar c,art7uantities are re&resented onl) b) t,e ,eig,t of t,e bar and its idt, signifiesnot,ingG ,ereas in a ,istogramG t,e area of t,e bar re&resents t,e 7uantit). InadditionG as a t,umb ruleG a ,istogram s,ould ,ave not less t,an % and not moret,an 1! bars.

: r e 7 u e n c )

/eadings

1re uen4" 5isto+r #

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I0, CONTROL C5ARTS . TOOL 1OR SPC

Dr. <alter S,e ,art of t,e ;ell AaboratoriesG in t,e 1+2 FsG first made t,e distinctionbet een controlled 8Common Causes9 and uncontrolled variation 8S&ecial Causes9Gand develo&ed a sim&le but &o erful tool to d)namicall) se&arate t,e t o J t,econtrol c,art. Control c,arts is one of t,e seven C Tools and are a sim&le andeffective tool to ac,ieve statistical control it &rovides a common language forcommunications about t,e &erformance of a &rocess. Control c,arts effectivel)direct attention to ard s&ecial causes of variation ,en t,e) a&&ear and reflect t,e

e tent of common cause variation t,at must be reduced b) management action.Several t)&es of control c,arts ,ave been develo&edG to anal)'e bot, variables andattributes. =o everG all control c,arts ,ave t,e same t o basic usesG t,e) areL

• $ judgmentG to give evidence ,et,er a &rocess ,as been o&erating in a stateof statistical controlG and to signal t,e &resence of s&ecial causes of variationso t,at corrective action can be ta en.

• $n o&erationG to maintain t,e state of statistical controlG b) e tending t,econtrol limits as a basis for realJtime decisions.

Process im&rovement using control c,arts is an iterative &rocedureG re&eating t,efundamental &,ases of collectionG control and ca&abilit). :irstG data are gat,eredaccording to a careful &lan t,enG t,ese data are used to calculate control limitsG

,ic, are t,e basis of inter&reting t,e data for statistical control ,en t,e &rocess isin statistical controlG it can be inter&reted for &rocess ca&abilit). To monitorim&rovements in control and ca&abilit)G t,e c)cle begins againG as more data aregat,eredG inter&retedG and used as t,e basis for action.

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customer needs. T,e &rocess itself must be investigatedG and managementaction must be ta en to im&rove t,e s)stem.

:or continuing &rocess im&rovementG re&eat t,ese t,ree &,ases. -at,er more dataas a&&ro&riate or to reduce &rocess variation b) o&erating t,e &rocess instatistical control and continuall) im&roving its ca&abilit).

Benefits of Control Ch rts .Pro&erl) usedG control c,arts canG• Control c,arts are sim&le and effective tools to ac,ieve statistical control. T,e)

can be maintained b) an o&erator. T,e) &rovide t,e reliable information to t,e&eo&le ,o are closest to t,e &rocess.

• =el& t,e &rocess &erform consistentl)G &redictabl)G for 7ualit) and costL&erformance to s&ecification of a statisticall) controlled &rocess is &redictableGt,us allo ing a consistentG stable and reliable 7ualit) levels.

• $fter a &rocess is in statistical controlG its &erformance can be furt,er im&roved toreduce variation. T,e e &ected effects of &ro&osed im&rovements can beantici&ated. Suc, &rocess im&rovementsG

• increase &ercentage of out&ut t,at meets t,e customere &ectations8 im&rove 7ualit)9• reduce t,e re or Qscra& 8cost &er unit9• increase t,e total )ield of acce&table out&ut t,roug, t,e

&rocess8im&rove effective ca&acit)9• Control c,arts &rovide a common language for communications about

&erformance of a &rocess.T,e common language is ver) critical because &rocess is in 2 to 3 s,iftsG

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need be c,ec ed to get total information about t,e &rocessG so in some casestotal ins&ection costs can be lo er.

4. Continuous ualit) Im&rovementL <it, variables dataG &erformance of a &rocessis anal)'ed even if all individual values are it,in t,e s&ecification limits t,is isim&ortant in see ing neverJending im&rovement.

#ariable c,arts can e &lain &rocess data in terms of bot, its s&read 8&ieceJtoJ&iecevariabilit)9 and its location 8&rocess average9. ;ecause of t,isG control c,arts forvariables are usuall) &re&ared and anal)'ed in &airs J one c,art for location andanot,er for s&read.T,e most commonl) used &air are t,e K / C,arts J K is t,e average of t,evalues in small subgrou&s J a measure of location / is t,e range of values it,ineac, subJgrou& 8,ig,est minus lo est9 J a measure of s&read.Pre&aration for se of #ariables Control C,artsL• >stablis, an environment suitable for action L T,e 0anagement must be

&re&ared and s,ould create a res&onsive environmentG b) &roviding training to&eo&leG resources for ualit) Im&rovementG etc.G before introducing an) Statistical

ualit) Im&rovement met,od• Define t,e &rocessL nderstand t,e &rocess in terms of its relations,i& to ot,er

o&erationsQusers bot, u&stream and do nstreamG and in terms of elements 8manGmac,ineG materialG met,odG environment9 t,at affect it at eac, stage. T,is is bestdone b) &eo&le ,o ,ave enoug, &rocess no ledgeable E use of tec,ni7uessuc, as causeJandJeffect diagram ,el& ma e t,ese relations,i&s visible.

• Determine c,aracteristics to be managedL Concentrate on vital fec,aracteristics 8t,ose t,at are most &romising for &rocess im&rovement9 E use oftec,ni7ues suc, as Pareto &rinci&le ma es &rioriti'ation eas)9. During &rioriti'ation

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controlled stud) it, no n in&ut materialsG constant control settingsG etc. 0aintaina &rocess log it, all relevant events suc, as tool c,angesG ne ra material lotsG

etc.G so t,at subse7uent &roblem anal)sis becomes eas).• Define /eaction PlansL S&ecif) reactions to situations involving abnormal or&eculiar &atterns of &rocess be,avior during t,e stud). Provisions s,ould also bemade for documenting.

CONTROL C5ARTS 1OR ATTRIBUTES $lt,oug, control c,arts are most often t,oug,t of in terms of variablesG versions,ave also been develo&ed for attributes. $ttributeJt)&e data ,ave onl) t o values8e.g.G conforming Q nonJconformingG &ass Q failG go Q noJgoG &resent Q absentG etc.9 butt,e) can be counted for recording and anal)sis &ur&ose. > am&les include t,econtinuit) of an electrical circuitG or number of incidences of runs Q scratc,es in a&ainted surface. > am&les also include c,aracteristics t,at are measurableG but

,ere t,e results are recorded in a sim&le )es Q no fas,ionG suc, as t,econformance of a s,aft diameter ,en measured on a go Q noJgo gageG t,eacce&tabilit) of door margins to a visual or gage c,ec G or onJtime deliver)&erformance. Control c,arts for attributes are im&ortant for t,e follo ing reasonsL1. $ttributeJt)&e situations e ist in an) tec,nical or administrative &rocessG so

attribute anal)sis tec,ni7ues are useful in man) a&&lications. T,e mostsignificant difficult) is to develo& &recise o&erational definitions of ,at is nonJconforming.

2. $ttributeJt)&e data are alread) available in man) situations J ,erever t,ere aree isting ins&ectionsG riteJu&s for re&airG sorts of rejected materialG etc. In t,esecasesG no additional data collection e &ense is involvedG just t,e effort ofconverting t,e data to control c,art form.

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T,e follo ing table illustrates various situations and t,e t)&e of attribute c,artem&lo)edG

Situ tion Out4o#e Control 4h rtTransmission E lea s Q does not lea P c,art for Pro&ortion of units nonJ

conformingN& c,art for number of units nonJconforming

=eadlam& E glo s Q does not glo=ole diameter E undersi'e Q oversi'ePart deliver) E inJtime Q not inJtimePart deliver) E com&lete Q notcom&leteNumber of bubbles in a inds,ieldframe

C c,art for number of nonJconformities&er ins&ection unit 8units of e7ual si'e9

c,art for number of nonJconformities&er ins&ection units 8units not of e7ualsi'e9

Number of &aint rundo ns on door Number of &atc,) coats on bonnetNumber of errors in an invoice

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The t :le :elo? :riefs so#e of the Ch rt T"$es

/ t Ch rt Me nin+ S #$le SiDe & 1e tures

V r i

: l e s

/

t H Indi%idu lMe sure#ent

S #$le SiDe @n (Ad% nt +es . Re uires less d t/is d% nt +es . Rel ti%el" insensiti%e nd$ro%ides $oor esti# te of % ri tion

H S #$leA%er +e

S #$le SiDe @n 2 3Ad% nt +es . Sensiti%e to s$e4i l 4 uses/is d% nt +es . Sensiti%e to @nor# l outliersin the d t

R S #$le R n+e S #$le SiDe @n 2 3Ad% nt +es . Sensiti%e to s$e4i l 4 uses/is d% nt +es . Loses res$onsi%eness ?ith

l r+e s #$le siDes8 hi+hl" :i sed #e sure of$ro4ess % ri tion

s S #$leSt nd rd/e%i tion

S #$le SiDe @n rel ti%el" s# ll8 u$to >Ad% nt +es . Sensiti%e to s$e4i l 4 uses &l r+er s #$les8 $ro%ides un:i sed #e sure of$ro4ess % ri tion8 e sil" 4o#$uterised/is d% nt +es . Need to 4 l4ul te @s nd is

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The t :le :elo? :riefs so#e of the Ch rt T"$es ;4ontdJ0<

/ t Ch rt Me nin+ S #$le SiDe & 1e tures

A t t r i : u

t e s

/

t $ Pro&ortions notconforming

Sam&le Si'e 5n6 ma) var)G usuall) large $dvantages L nl) re7uires counts data 8nomeasurements9Disadvantages L /e7uires large sam&lesG needs

clear definition of s&ecifications 8acce&table notacce&table9G needs rigid enforcement ofacce&tance standards to be effectiveGinter&retation difficult ,en different t)&es ofdefects define a nonJconformanceG somecalculations ma) be re7uired

n$ Number notconforming

Sam&le Si'e 5n6 is constantG $dvantages L nl) re7uires counts data 8nomeasurements9Disadvantages L Needs clear definition ofs&ecificationsG some calculations ma) be re7uired

4 Number of nonJconformities

Sam&le Si'e 5n6 is constantG $dvantages L nl) re7uires counts data 8nomeasurements9

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SECTION II . CONTROL C5ARTS

II0( GUI/ELINES 1OR SELECTION O1 APPROPRIATE C5ART

V rious t"$es of Control Ch rts re e#$lo"ed : sed on the t"$e of d t+ener ted0 The 4h rt :elo? $ro%ides infor# tion :out sele4tion4riteri of 4ontrol 4h rts used in % rious situ tions0

!amplesH & R8 H & s . $ series of sam&lesG usuall) four to si items are measured to getvalues of selected &arameter.E ample! T,e best e am&le can be journal diameter of cran s,aft J D grinding

H & M R

n I (

H & R o r H & s

n L (

" s n # $ o r n % $

V r i : l e

u

U n i t

4

P o r t i o n

& n i t o r ' o r t io n

n I (

n $

C o n s t n t

$

V r " i n +

( o n s t a n t o r ) a r y i n g

n L (

" s n # $ o r n % $

A t t r i : u t e

T " $ e o f / t

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Ch rt for sele4tion of Control Ch rt

C n su: +rou$%er +es :e

4on%enientl"4o#$uted

n$ or $Ch rt

4 or uCh rt

/eter#ine the Ch r 4teristi4 to :e 4h rted

Is theinterest in

non4onfor#in+units K i0e08

: d$ rts

Are thed t

V ri :le

Is theInterest in

non 4onfor#itiesunits K i0e08

dis4re$ n4ies$er $ rts

$Ch rt

uCh rt

Is theS #$le

siDe4onst nt

Is theS #$le

siDe4onst nt

Is itho#o+e

neous in n tureor not 4ondu4i%e

to su: +rou$s #$lin+ K e0+084he#i4 l : th8

$ int: t4h8et40

Medi nCh rt

H>S

N N

H>S H>S

N N

H>S

N N

H>S

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II0- CONTROL C5ARTS 1OR VARIABLES H ;AVERAGE< & R ;RANGE<

(0 G ther / t $ dataJgat,ering &lan must be develo&ed and used as t,e basis for collectingGrecording and &lotting t,e data on a c,art. T,e data 8measurements of a&articular c,aracteristic of t,e &rocess out&ut9 are re&orted in small subgrou&s ofconstant si'eG usuall) including from 2 to ! consecutive &iecesG it, subgrou&sta en &eriodicall)G 8e.g.G once ever) 1! minutesG t ice &er s,iftG etc.9

$. Si'eG :re7uenc) and Number of Subgrou&si. Subgrou& Si'e L :or effectiveness and efficienc) of t,e control c,artsG

determination of rational subgrou&s is t,e e) ste&.&&ortunities for variation among t,e units it,in a subgrou& are

minimal. <it, &ieceJtoJ&iece variabilit) it,in a subgrou&G an) unusualvariation bet een subgrou&s ould reflect c,anges in t,e &rocess t,ats,ould be investigated for a&&ro&riate action.:or an initial stud) of a &rocess 4 to ! consecutivel) &roduced &iecesfrom a single &rocess streamG under ver) similar &roduction conditionsover a ver) s,ort time interval it, no ot,er s)stematic relations,i& toeac, ot,er 8onl) a single toolG ,eadG die cavit)G etc.9 s,ould be c,osen.T,us variation it,in eac, subgrou& ould &rimaril) reflect commoncauses. <,en t,ese conditions are not metG t,e resulting control c,artma) not effectivel) discriminate s&ecial causes of variationG or ma)e ,ibit nonJrandom &atterns.ii. Subgrou& :re7uenc)L :or detecting c,anges in t,e &rocess over timeG

subgrou&s s,ould be collected often enoug,G and at a&&ro&riate timesGt,at t,e) can reflect t,e &otential o&&ortunities for c,ange it, res&ect

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substantiall) ,ig,er or lo er t,an t,e ot,ers to confirm t,at t,e calculationsand &lots are correct.

-0 C l4ul te Control Li#itsControl limits for t,e range 8/9 c,art are develo&ed firstG t,en t,ose for t,e c,artfor averages 8 K 9. T,e ste&s involved includeG

$. Calculate t,e $verage /ange 8 R 9 and t,e Process $verage 8 H 9 asbelo L

( )

9

R 000 R R R R 9)-( ++++=

)9

H 000 H H H H 9)-( ++++=

,ereG 5 6 is t,e number of subgrou&sG (R G -R G )R G 9R (H G -H G )H G9H are t,e corres&onding ranges averages of t,e 1stG 2ndG 3rdG t,

subgrou&sG res&ectivel).;. Calculate t,e Control Aimits L Control limits are calculated to s,o t,e e tent

b) ,ic, t,e subgrou& averages and ranges ould var) if onl) commoncauses of variation ere &resent. T,e Control Aimits are calculated using t,ela s of &robabilit) t,at ,ig,l) im&robable causes of variations are &resumedto be not due to random causesO. <,en t,e variation e ceeds t,e statisticalcontrol limitsG it is a signal t,at a s&ecial cause ,as entered t,e &rocess.T,e) are based on t,e subgrou& sam&le si'e and t,e amount of it,inJsubgrou& variabilit) reflected in t,e ranges.Calculate t,e control limits as belo L

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limits 8less t,an 1M of t,e time for rangesG and onl) .2(M of t,e time foraveragesG for t,e limits calculated above9.

T,eH

and / c,art are anal)'ed se&aratel)G t,e / c,art is anal)'ed firstfollo ed b) t,e H c,artG but com&arison of &atterns bet een t,e t o c,artsma) sometimes give added insig,t into s&ecial causes affecting t,e &rocess.T,e ste&s involved in Control C,art $nal)sis are as belo L

$. $nal)se t,e data &lots on t,e C,artT,e data &oints are com&ared it, t,e res&ective control limitsG for &oints outof control or for unusual &atterns or trends. :or more details see Inter&retationof Control C,arts.

Points be)ond t,e Control AimitsL It is &rimar) evidence of nonJcontrol at t,at&ointG and it, variation onl) due to common causes suc, &oints ould bever) rare. $n) &oints be)ond a control limit is t,e signal for immediateanal)sis of t,e o&eration for t,e s&ecial causeG and mar suc, &oints forfurt,er investigation and corrective action. /efer to t,e follo ing table for&ossible causes 8for more details refer to $&&endi KHB9 L

Condition H R

Pointsabove

CA

• 0iscalculation

• Incorrect Plotting

• C,ange in 0easurement S)stem E differentins&ectorG different gaugeG incorrect readingG etc.

Pointsbelo

• 0iscalculation

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is of great ,el& in terms of ,aving fres, evidence for diagnosis of t,ecause of concern and minimi'ing t,e &roduction of nonJconforming

out&ut. >ven t,e a&&earance of a single &oint be)ond t,e controllimits is reason to begin an immediate anal)sis of t,e &rocess.ProblemJsolving is often t,e most difficultG but Statistical in&ut fromt,e control c,art can be an a&&ro&riate starting &oint along it, ot,ersim&le tools suc, as Pareto c,artsG cause and effect diagrams 8see

$&&endi )' for more details9. Process im&rovement is largel)associated it, t,e &rocess design t,e &eo&le ,o are involved

it, it.

b. /ecalculate Control Aimits<,en conducting an initial &rocess stud) 8or a reassessment of&rocess ca&abilit)9G e clude all subgrou&s affected b) t,e s&ecialcauses 8e clude t,e effects of outJofJcontrol &eriods9 t,at ,ave beencorrected. T,e e clusion of subgrou&s re&resenting unstableconditions is not just t,ro ing a a) bad data. /at,erG b) e cludingt,e &oints affected b) no n s&ecial causesG one ,as a betterestimate of t,e bac ground level of variation due to common causes.

Confirm t,at all &lots s,o control ,en com&ared to t,e ne limitsGre&eating t,e identification Q correction Q recalculation se7uence ifnecessar).If an) subgrou&s ere dro&&ed from t,e / c,art because of identifieds&ecial causesG t,e) s,ould also be e cluded from t,e K c,art andviceJversa. T,e revised R H s,ould be used to recalculate t,etrial control limits for averages as belo G

×==

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limitsG e tend t,e limits for ongoing control of t,e &rocess 8t,eo&erator and local su&ervision res&onding to signs of outJofJcontrol

conditions on eit,er t,eH

or / c,art it, &rom&t action9."ote ! change in the subgroup sample si/e *ould affect thee pected a%erage range and the control limits for both ranges anda%erages. Such a situation could occur if it *as decided to ta+esmaller samples more fre0uently so as to detect large process shiftsmore 0uic+ly *ithout increasing the total number of pieces sampled

per day.To adjust central lines and control limits for a ne subgrou& sam&le

si'eG t,e follo ing ste&s s,ould be ta enL• >stimate t,e &rocess standard deviation 8t,e estimate is s,o n as σ ̂

8&ronounced as sigma ,at9 ,at 9. sing t,e e isting sam&le si'ecalculate L

-dR

N =ˆ

,ere R is t,e average of t,e subJgrou& ranges 8for &eriods it, t,e

ranges in control9 and -d is a constant var)ing b) sam&le si'e8$&&endi E KHB9• sing t,e tabled factors for d 2G D3G D4G and $ 2 based on t,eG ne

sam&le si'eG calculate t,e ne range and control limitsG

-ne? d N R ×= ˆ

ne?2;on+oin+<Ron+oin+ R / UCL ;R<Li#itControlU$$er ×==

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Since t,e abilit) to inter&ret eit,er t,e subgrou& ranges or subgrou& averagesde&ends on t,e estimate of &iece to &iece variabilit)G t,e / c,art is anal)'ed

first. T,e data &oints are com&ared it, t,e control limitsG for &oints out ofcontrol or unusual &atterns or trends 0C. T,e inter&retation s,ould be done b) considering t,ree im&ortant as&ects

• /uns• Trend• Periodicit)

i. $nal)sis of data &lots on t,e /ange c,arta. Points be)ond t,e Control Aimits LT,e &resence of one or more &oints be)ond eit,er control limit is&rimar) evidence of nonJ control at t,at &oint. Since &oints be)ondcontrol limits ould be rare if onl) variation from common causes

ere &resentG e can &redict t,at a s&ecial cause ,as accounted fort,e e treme value. $n) &oints be)ond a control limit is t,e signal forimmediate anal)sis of t,e o&eration for t,e s&ecial causeG and marsuc, &oints for furt,er investigation and corrective action.b. Patterns and trends it,in t,e control limitsT,e &resence of unusual &atterns or trends even ,en all ranges are

it,in t,e control limitsG can be evidence of nonJcontrol or c,anges in&rocess s&read during t,e &eriod of &attern or trend.c. /unsT,e follo ing are t,e signs of a &rocess s,ift or needs correction

• ( consecutive &oints or 1 out of 22 or 12 out of 14 consecutivereadings fall on t,e same side of t,e central line.

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is of great ,el& in terms of ,aving fres, evidence for diagnosis of t,ecause of concern and minimi'ing t,e &roduction of nonJconforming

out&ut. >ven t,e a&&earance of a single &oint be)ond t,e controllimits is reason to begin an immediate anal)sis of t,e &rocess.ProblemJsolving is often t,e most difficultG but statistical in&ut fromt,e control c,art can be an a&&ro&riate starting &oint along it, ot,ersim&le tools suc, as Pareto c,artsG cause and effect diagrams.Process im&rovement is largel) associated it, t,e &rocess design t,e &eo&le ,o are involved it, it.

ii. /ecalculate Control Aimits

<,en conducting an initial &rocess stud) 8or a reassessment of&rocess ca&abilit)9G e clude all subgrou&s affected b) t,e s&ecialcauses 8e clude t,e effects of outJofJcontrol &eriods9 t,at ,ave beencorrected. T,e e clusion of subgrou&s re&resenting unstableconditions is not just t,ro ing a a) bad data. /at,erG b) e cludingt,e &oints affected b) no n s&ecial causesG one ,as a betterestimate of t,e bac ground level of variation due to common causes.Confirm t,at all &lots s,o control ,en com&ared to t,e ne limitsG

re&eating t,e identification Q correction Q recalculation se7uence ifnecessar).If an) subgrou&s ere dro&&ed from t,e / c,art because of identifieds&ecial causesG t,e) s,ould also be e cluded from t,e H c,art andviceJversa.

iii. $nal)'e t,e data &lots on average c,art<,en t,e ranges are in statistical controlG t,e &rocess s&read E t,e

it,in sub grou& variation is considered to be stable. T,e averages

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d. bvious Non random &atternSimilar met,od as described for / c,art ,as to be follo ed.

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II0) CONTROL C5ARTS 1OR VARIABLES H ;AVERAGE< & s ;STAN/AR/

/EVIATION<

Ai e t,e H / c,artsG H s c,arts are develo&ed from variable &rocess out&utdataG and are al a)s used as a &air. /ange 8/9 c,arts are &articularl) effective dueto ease of calculations involvedG ,o everG /Jc,arts are relativel) efficient onl) forsmaller subgrou& si'es 8es&eciall) belo *9. T,e Sam&le Standard Deviation 8s9 issome ,at more efficient indicator of Process #ariabilit)G ,o everG it is more com&leto calculate and is less sensitive in detecting S&ecial Causes of #ariation t,at cause

a single value in a subgrou& t,at is unusual. T)&icall) sJc,arts are used instead of/Jc,arts ,enG

• Aarger sam&le si'es are re7uired and a&&ro&riate for more efficient measureof #ariation.

• Com&utation of Sam&le Standard Deviation 5s6 is facilitated it,calculatorsQcom&uters.

T,e ste&s involved are 8similar to t,ose mentioned in H R9 C,arts as follo sG(0 G ther / t

$ dataJgat,ering &lan must be develo&ed and used as t,e basis for collectingGrecording and &lotting t,e data on a c,art. T,e data 8measurements of a&articular c,aracteristic of t,e &rocess out&ut9 are re&orted in subgrou&s ofconstant si'eG usuall) including from * to 1 consecutive &iecesG it, subgrou&sta en &eriodicall)G 8e.g.G once ever) 1! minutesG t ice &er s,iftG etc.9

$. Si'eG :re7uenc) and Number of Subgrou&si. Subgrou& Si'e L :or effectiveness and efficienc) of t,e control c,artsG

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demonstrates stabilit) 8or as &rocess im&rovements are made9G t,e timeJ&eriod bet een subgrou&s can be increased.

iii. Number of Subgrou&sL >noug, subgrou&s s,ould be gat,ered toassure t,at t,e major sources of variation ,ave ,ad an o&&ortunit) toa&&ear. Statisticall)G 2! or more subgrou&s containing about 2 or moreindividual readings give a good test for stabilit) andG if stableG goodestimates of t,e &rocess location and variation.In some casesG e isting data ma) be available ,ic, could acceleratet,is first &,ase of t,e stud). =o everG t,e) s,ould be used onl) if t,e)are recent and if t,e basis for establis,ing subgrou&s is clearl)

understood.iv. If t,e ra data is voluminousG record data on a se&arate data s,eetGit, onl) eac, subgrou&6s H and s a&&earing on t,e data s,eet.

;. Set & Control C,arts and /ecord /a Data L:ollo t,e follo ingG

i. In case of voluminous dataG collect data 8readings9 using t,e 5DataCollection S,eet6ii. Com&lete t,e =eader Information of t,e Data Collection S,eetiii. /ecord data on t,e Data Collection S,eetiv. Com&lete t,e =eader Information of t,e Control C,art.v. Include average 8 H 9G sam&le standard deviation 8s9 and dateQtime8or ot,er identification of t,e subgrou&9G from t,e data s,eetG in t,ebottom bloc .vi. Dra t,e H c,art above t,e s c,artvii. #alues of H and s ill be t,e vertical scales

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( )2(n

HH s i∑ −=

or

( )(nHnH

s-i∑ −

=2

or

( )(nHn 00000000H0000000000 H HH

s--

n-)

--

-( ++++=

<,ere 5K 1 G 5K26G ". 5Kn6 are t,e individual values it,in t,e subgrou& and 5n6 ist,e subgrou& sam&le si'e.

D. Select Scales for t,e Control C,arts:or a better fitG t,e general guidelines for determining scales are as underG

i. :or t,e H c,art do t,e follo ing L• Calculate 8 H ,ig,est E H lo est9 5$6• Calculate 8 H lo est E $9 5;6• Calculate 8 H ,ig,est $9 5C6• Calculate 8C E ;9 5D6• :ind t,e number of increments on t,e c,art 5n6• Divide 5D6 b) 5n6 8D Q n9 5>6

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Control limits for t,e sam&le standard deviation 8s9 c,art are develo&ed firstG t,ent,ose for t,e c,art for averages 8 H 9. T,e ste&s involved includeG

$. Calculate t,e $verage Sam&le Standard Deviation 8 s 9 and t,e Process $verage 8 H 9 as belo L

( )9

s 000 s s s s 9)-( ++++=

)9

H 000 H H H H 9)-( ++++=

,ereG 5 6 is t,e number of subgrou&sG (s G -s G )s G 9s (H G -H G )H G9H are t,e corres&onding sam&le standard deviations averages of t,e 1stG

2ndG 3rdG t, subgrou&sG res&ectivel).;. Calculate t,e Control AimitsL Control limits are calculated to s,o t,e e tent

b) ,ic, t,e subgrou& averages and sam&le standard deviations ould var)if onl) common causes of variation ere &resent. T,e Control Aimits arecalculated using t,e la s of &robabilit) t,at ,ig,l) im&robable causes of

variations are &resumed to be not due to random causesO. <,en t,evariation e ceeds t,e statistical control limitsG it is a signal t,at a s&ecialcause ,as entered t,e &rocess. T,e) are based on t,e subgrou& sam&lesi'e and t,e amount of it,inJsubgrou& variabilit) reflected in t,e sam&lestandard deviations.

Calculate t,e control limits as belo Ls B UCL ;s<Li#itControlU$$er 2s ×==

s B LCL ;s<Li#itControlLo?er )s ×==

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seldom go be)ond t,e control limits 8less t,an 1M of t,e time for sam&lestandard deviationsG and onl) .2(M of t,e time for averagesG for t,e limitscalculated above9.T,e H and s c,art are anal)sed se&aratel)G t,e s c,art is anal)sed firstfollo ed b) t,e H c,artG but com&arison of &atterns bet een t,e t o c,artsma) sometimes give added insig,t into s&ecial causes affecting t,e &rocess.T,e ste&s involved in Control C,art $nal)sis are as belo L

$. $nal)se t,e data &lots on t,e C,artT,e data &oints are com&ared it, t,e res&ective control limitsG for &oints outof control or for unusual &atterns or trends. :or more details see Inter&retation

of Control C,arts.Points be)ond t,e Control AimitsL It is &rimar) evidence of nonJcontrol at t,at&ointG and it, variation onl) due to common causes suc, &oints ould bever) rare. $n) &oints be)ond a control limit is t,e signal for immediateanal)sis of t,e o&eration for t,e s&ecial causeG and mar suc, &oints forfurt,er investigation and corrective action. /efer to t,e follo ing table for&ossible causes 8for more details refer to $&&endi KHB9L

Condition H s

Points:o%e

UCL

• Mis4 l4ul tion

• In4orre4t Plottin+

• Ch n+e in Me sure#ent S"ste# K differentins$e4tor8 different + u+e8 in4orre4t re din+8 et40

Points • Mis4 l4ul tion

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condition began and ,o long it continued. $ &ro&erl) maintained logis of great ,el& in terms of ,aving fres, evidence for diagnosis of t,ecause of concern and minimising t,e &roduction of nonconformingout&ut. >ven t,e a&&earance of a single &oint be)ond t,e controllimits is reason to begin an immediate anal)sis of t,e &rocess.ProblemJsolving is often t,e most difficultG but Statistical in&ut fromt,e control c,art can be an a&&ro&riate starting &oint along it, ot,ersim&le tools suc, as Pareto c,artsG Cause and >ffect diagrams 8see

$&&endi )' for more details9. Process im&rovement is largel)associated it, t,e &rocess design t,e &eo&le ,o are involved

it, it.b. /ecalculate Control Aimits<,en conducting an initial &rocess stud) 8or a reassessment of&rocess ca&abilit)9G e clude all subgrou&s affected b) t,e s&ecialcauses 8e clude t,e effects of outJofJcontrol &eriods9 t,at ,ave beencorrected. T,e e clusion of subgrou&s re&resenting unstableconditions is not just t,ro ing a a) bad data. /at,erG b) e cludingt,e &oints affected b) no n s&ecial causesG one ,as a better

estimate of t,e bac ground level of variation due to common causes.Confirm t,at all &lots s,o control ,en com&ared to t,e ne limitsGre&eating t,e identification Q correction Q recalculation se7uence ifnecessar).If an) subgrou&s ere dro&&ed from t,e s c,art because of identifieds&ecial causesG t,e) s,ould also be e cluded from t,e K c,art andviceJversa. T,e revised s H s,ould be used to recalculate t,etrial control limits for averages as belo G

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c. > tend Control Aimits for ngoing ControlL nce t,e initial 8or,istorical9 data are consistentl) contained it,in t,e trial controllimitsG e tend t,e limits for ongoing control of t,e &rocess 8t,eo&erator and local su&ervision res&onding to signs of outJofJcontrolconditions on eit,er t,e H or s c,art it, &rom&t action9."ote! change in the subgroup sample si/e *ould affect thee pected a%erage sample standard de%iation and the control limitsfor both sample standard de%iation and a%erages. Such a situationcould occur if it *as decided to ta+e smaller samples morefre0uently so as to detect large process shifts more 0uic+ly *ithout

increasing the total number of pieces sampled per day.To adjust central lines and control limits for a ne subgrou& sam&lesi'eG t,e follo ing ste&s s,ould be ta enL

• >stimate t,e &rocess standard deviation 8t,e estimate iss,o n as σ ̂ 8&ronounced as sigma ,at9. sing t,e e istingsam&le si'e calculate L

24s

=σˆ

,ere s is t,e average of t,e subJgrou& sam&le standarddeviations 8for &eriods it, t,e standard deviations undercontrol9 and 24 is a constant var)ing b) sam&le si'eG from 2 to1 . 8$&&endi E KHB9• sing t,e tabled factors for c 4G ;4G ;3G and $ 3 based on t,eG

ne sam&le si'eG calculate t,e ne standard deviation andcontrol limitsG

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In all suc, cases ,ere grou& si'e is oneG 0/ 80oving /ange E difference bet eent,e value and t,e one immediatel) &receding it9 is calculated instead of range usedfor sam&les it, grou& si'e of t o or more.

T,e follo ing are t,e ste&s for ma ing t,e c,artGSte$ (G ther d tT,e first ste& is to note t,e observations of all t,e sam&les. -enerall) minimumsam&le si'e is 2!. >ac, grou& ill ,ave onl) one observation. ne s,ould becareful in giving t,e serial numbersG because t,e moving range needs to be

calculated from consecutive observations.Ste$ -T,e first ste& is to calculate t,e 0oving /ange 8 MR9. It must be noted t,at e getvalues of MR one number less t,an t,e total observationsG as t,ere is no MRcorres&onding to t,e first value.

If H( 8 H - 8 H) 8 JJ H n re the o:ser% tions8 then the Mo%in+ R n+e ;MR< is4 l4ul ted s follo?s8

<Re din+Re din+; R n+e Mo%in+ <(n;nn =T,usG

Mo%in+ R n+e ( ;MR ( < Nil8as t,ere is no t, readingMo%in+ R n+e - ;MR - < H- H(

Mo%in+ R n+e ) ;MR ) < H) H- 00

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=

+=

=

<(0()MR

; F )HLCL; HLi#it4ontrolLo?er

<(0()MR ; F )HUCL; HLi#it4ontrolU$$er

H lineCenter

F

F

)

)

Calculate t,e control limits of MR c,art.

'F MRLCL;MRfor Li#it4ontrolLo?er

)0-=F MRUCL;MRfor Li#it4ontrolU$$er

MR lineCenter

MR

MR

=

=

=

)

)

Ste$ 3Plot bot, t,e c,artsL It s,ould be noted t,at t,e &rocedure to set u& an H MR c,art

is almost same as t,at for H R c,artG t,e onl) difference is t,at e use H and MRinstead of H and R .Ste$ *Inter&ret t,e c,artL T,e basic rules of inter&retation remain same as t,osementioned in H R c,art.

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In some casesG e isting data ma) be available ,ic, could accelerate t,is first&,ase of t,e stud). =o everG t,e) s,ould be used onl) if t,e) are recent and if t,ebasis for establis,ing subgrou&s is clearl) understood.Ste$ -. Plot the d t

W nl) single line gra&, is &lottedG set t,e scale to include t,e larger of a9 t,e&roduct s&ecification tolerance &lus an allo ance for out of s&ecification readings orb9 1J1Q2 to 2 times t,e difference bet een t,e ,ig,est and lo est individualmeasurements. T,e gauge being used s,ould divide t,e &roduct tolerance intoatleast 2 increments and t,e gra&, scale s,ould agree it, t,e gauge.W Plot individual measurements of eac, sub grou& in a vertical line. Circle t,emedian of eac, subgrou& 8t,e middle valueG if sam&le si'e is even numberG t,emedian ill be mid a) bet een t,e inner &oints9 To aid in inter&reting trendsGconnect t,e subgrou& medians b) a line.W >nter eac, sub grou& 5s median and range in t,e data table.Ste$ )0 C l4ul te Control Li#its

• :ind t,e average of t,e subgrou& medians and dra t,is as t,e central lineon t,e c,art record t,is as K .

• :ind t,e average of t,e ranges record t,is as / .• Calculate t,e u&&er and lo er control limits for ranges and medians• ;UCL R8 LCLR8 UCL K

~ 8 LCL K~ <.

UCL R /2 R

LCL R /) R

UCL H~ RAH +

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Ste$ 20 Inter$ret for Pro4ess Control

• Com&are t,e CA / and ACA / it, eac, calculated range. $lternativel)G mar

t,e edge of an inde card it, t,e &oints corres&onding to t,e control limitsfor rangesG and com&are t,ese mar s it, t,e distance bet een t,e ,ig,estand lo est value in eac, subgrou&. Dra a narro vertical bo to enclosean) subgrou& it, e cessive range.

• 0ar an) subgrou& median t,at is be)ond t,e median control limitsG andnote t,e s&read of medians it,in t,e control limits 82Q3 of &oints it,inmiddle t,ird of limits9 or t,e e istence of &atterns or trends.

• Ta e a&&ro&riate &rocess action for s&ecial causes affecting t,e ranges ormedians.

Ste$ 30 Inter$ret for Pro4ess C $ :ilit"

• >stimate t,e &rocess standard deviationsL

2

ˆd

/=σ

,ere/

is t,e average of t,e sam&le ranges 8for &eriods it, t,e rangeunder control9 and d2 is a constant var)ing b) sam&le si'eG it, values forsam&le si'es from 2 to 1 s,o n in t,e follo ing tableL

n - ) 2 3 * = > , ('d - (0() (0*, -0'* -0)) -03) -0=' -0>3 -0,= )0'>

• If t,e &rocess ,as a normal distributionG t,is estimate of σ can be useddirectl) in assessing &rocess ca&abilit)G as long as t,e medians and ranges

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/is d% nt +es of Medi n 4h rts

Periodicall) t,e median is more a variable estimate of t,e &rocess average t,an t,e

mean. <ilder control limits are t,erefore necessar) ,ic, ma) give rise to undercontrol of t,e &rocess.

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SECTION II * . CONTROL C5ARTS 1OR VARIABLES K IN/IVI/UALS ;H<

In some casesG it is necessar) for &rocess control to be based on individualreadingsG rat,er t,an subgrou&s. T,is ould t)&icall) occur ,en t,e measurementsare e &ensive 8e.g.G a destructive test9G or ,en t,e out&ut at an) &oint in time isrelativel) ,omogenous 8e.g.G t,e &= of a c,emical solution9. In t,ese casesG controlc,arts for individuals can be constructed as described belo . :our cautions s,ouldbe notedG ,o everL

• C,arts for individuals are not as sensitive to &rocess c,anges as H and /c,arts.• Care must be ta en in inter&retation of c,arts for individuals if t,e &rocess

distribution is not s)mmetrical.• C,arts for individuals do not isolate t,e &ieceJtoJ&iece re&eatabilit) of t,e

&rocess. In man) a&&licationsG t,ereforeG it ma) be better to use a conventional X and / c,art it, small subgrou& sam&le si'es 82 to 49 even if t,is re7uires a longer&eriod bet een subgrou&s.

• Since t,ere is onl) one individual item &er subgrou&G values of X and σ ̂ can,ave substantial variabilit) 8even if t,e &rocess is stable9 until t,e number ofsubgrou&s is 1 or more.

T,e details of instructions for c,arts for individuals are some ,at similar to t,ose forH and / c,arts e ce&tions are noted belo L

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Ste$ -0 C l4ul te Control Li#its

Calculate and &lot t,e &rocess average 8 t,e sum of individual readingsG divided b)

t,e number of readings b) conventionG labeled X 9G and calculate t,e averagerange 8 R 9 note t,at t,ere is one less range value 8/ 9 t,an t,e number of individualreadings 8K9.

• Calculate t,e control limitsL

REHLCL

REH UCL

R F/LCL

RF/UCL

-F

-F

)R

2R

−=

+=

=

=

<,ere R is t,e average moving rangeG H is t,e &rocess averageG and D4G D3 and>2 are constants t,at var) according to t,e sam&le si'e used in grou&ing t,e movingrangesG as s,o n in t,e follo ing &artial tableL

n 2 3 4 ! % ( * + 1D4 3.2( 2.!( 2.2* 2.11 2. 1.+2 1.*% 1.*2 1.(*

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2

ˆd

R=σ

,ere R is t,e average of t,e moving ranges and d 2 is a constantvar)ing b) sam&le si'eG as s,o n in t,e &artial table belo L

5n 2 3 4 ! % ( * + 15d2 1.13 1.%+ 2. % 2.33 2.!3 2.( 2.*! 2.+( 3. *

• If t,e &rocess ,as a normal distributionG t,is estimate of σ can be used

directl) in assessing &rocess ca&abilit)G as long as t,e &rocess is instatistical control.

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iii. T,ese data s,ould be recorded on a data form as t,e basis of initialanal)sis. In case if t,e most recent aut,entic ,istorical data areavailableG it ma) be used to accelerate t,is initial &,ase of t,e stud).

C. Set & Control C,arts and /ecord /a Datai. Com&lete t,e =eader Information of t,e Control C,art.ii. Include corres&onding data vi'. Sam&le or number of items ins&ected

8n9G Number of nonJconforming items found 8n&9G Pro&ortion nonJconforming 8&9 and DateQTime 8or ot,er identification of t,e subgrou&9in t,e bottom bloc .

iii. Subgrou& identification t,roug, time 8,ourG da)G etc.9 ill be t,e

,ori'ontal scaleiv. #erticall) align t,e corres&onding &ro&ortion 8or &ercent9 nonJconforming values 8&9

v.> tend t,e vertical scale from 'ero 8 . 9 to about 1.! to 2 times t,e,ig,est &ro&ortion nonJconforming noted in t,e initial data readings.

vi. Plot and connect t,e values of & for eac, subgrou&. It is usuall),el&ful to connect t,e &oints it, lines to ,el& visuali'e &atterns andtrends. Scan t,e &lot &oints to see if t,e) loo reasonable and no &oint

is substantiall) ,ig,er or lo er t,an t,e ot,ers to confirm t,at t,ecalculations and &lots are correct

Ste$ -0 C l4ul te Control Li#itsi. Calculate t,e Process $verage Pro&ortion NonJconforming 8 $ 9

:or t,e stud) &eriod of subgrou&sG calculate t,e average &ro&ortion nonJconforming L

n$0000 n$ n$ n$ 9)-( ++++

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• <,en $ is lo andQor n is smallG t,e ACA can sometimes be calculated asa negative number. In t,ese cases t,ere is no Ao er Control AimitG sinceeven a value of & for a &articular &eriod is it,in t,e limits of randomvariation.

• T,e Control Aimit calculations given above are a&&ro&riate ,en t,esubgrou& si'es are all e7ual 8as t,e) ould be in a controlled sam&lingsituation9. =o everG ,enever t,e sam&le si'e c,anges 8even for a singlesubgrou&9G t,e Control Aimits c,angeG and t,e uni7ue limits ould becalculated for eac, subgrou& ,aving a uni7ue sam&le si'e. :or all &ractical&ur&osesG Control Aimits calculated it, an average sam&le si'e 8 n 9 are

acce&table ,en t,e individual subgrou& si'es var) from t,e average b) nomore t,an ± 2!M 8t)&ical of actual &roduction volumes under relativel)stable conditions9.

• <,en subgrou& si'es var) b) more t,an ± 2!MG se&arate Control Aimits arere7uired for t,e &eriods it, &articularl) small or large sam&les. In all suc,casesG

• determine t,e range of sam&le si'es t,at ould var) from t,eaverage b) ± 2!M

• identif) all subgrou&s it, sam&le si'es t,at lie outside t,isrange.

• reJcalculate t,e &recise limits for t,ose &oints using t,e formulaG

( )n

$($) $ UCL ;$<Li#itControlU$$er $ +==

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Ste$ ). Inter$ret for Pro4ess ControlIdentif) an) evidence t,at t,e &rocess is no longer o&erating at t,e same level Jt,at it is out of control J and to ta e a&&ro&riate action. In case of Statisticall)Controlled &rocesses t,ere ould be no obvious trends or &atterns in t,e data.i. $nal)'e t,e data &lots on t,e C,art 8see also Control C,art Inter&retations9T,e data &oints are com&ared it, t,e res&ective control limitsG for &oints out ofcontrol or for unusual &atterns or trends. :or more details see Inter&retation ofControl C,arts.

• Points ;e)ond t,e Control Aimits J T,e &resence of one or more &ointsbe)ond eit,er control limit is evidence of instabilit) at t,at &oint. Since

&oints be)ond t,e control limits ould be ver) rare if t,e &rocess ere stableand onl) commonJcause variation ere &resentG e &resume t,at a s&ecialcause ,as accounted for t,e e treme value. T,e s&ecial cause ma) beeit,er unfavorable or favourable E eit,er bears immediate investigation.T,is is t,e &rimar) decision rule for action on an) control c,art. $n) &ointbe)ond t,e control limits s,ould be mar ed.

• $ &oint above t,e u&&er control limit 8,ig,er &ro&ortion nonJconforming9 isgenerall) a sign t,at L

• T,e control limit or &lot &oint are in errorG or • T,e &rocess &erformance ,as orsenedG eit,er at t,at &oint in

time or as &art of a trendG or• T,e measurement s)stem ,as c,anged 8e.g.G ins&ectorG gage9.

• $ &oint belo t,e lo er control limit 8lo er &ro&ortion nonJconforming9 isgenerall) a sign t,atL

• T,e control limit or &lot &oint are in errorG or

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• ( intervals in a ro t,at are consistentl) increasing 8e7ual orgreater t,an t,e &receding &oints9 or consistentl) decreasing.

In t,ese casesG t,e &oint t,at &rom&ts t,e decision s,ould bemar ed 8e.g.G t,e eig,t, &oint above t,e average9 it ma) be ,el&fulto e tend a reference line bac to t,e beginning of t,e run. T,eanal)sis s,ould consider t,e a&&ro imate time at ,ic, it a&&earst,at t,e trend or s,ift first began.

• /uns above t,e &rocess averageG or runs u&G generall) signif) t,at L• T,e &rocess &erformance ,as orsened J and ma) still be

orseningG or

• T,e measurement s)stem ,as c,anged.• /uns belo t,e &rocess averageG or runs do nG generall) signif) t,at L

• T,e &rocess &erformance ,as im&roved 8t,e causes s,ould bestudied for &ermanent incor&oration9G or

• T,e measurement s)stem ,as c,anged.N T>L <,en n $ is small 8belo !9G t,e li eli,ood of runs belo$ increasesG so a run lengt, of * or more could be necessar) to

signal a decrease in t,e &ro&ortion nonJconforming.• bvious Nonrandom PatternsL t,er distinct &atterns ma) indicate t,e&resence of s&ecial causes of variationG alt,oug, care must be ta en not tooverJinter&ret t,e data. $mong t,ese &atterns are trendsG c)clesG unusuals&read of &oints it,in t,e control limitsG and relations,i&s among values

it,in subgrou&s 8e.g.G if all nonJconforming items occur it,in t,e first fereadings ta en for t,e subgrou&9. ne test for unusual s&read is givenbelo L

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( )

re%ised

re%isedre%isedre%ised$;re%ised<re%ised

n

$($) $ UCL ;$<Li#itControlU$$er +==

( )

re%ised

re%isedre%isedre%ised$;re%ised<re%ised

n

$($) $ LCL ;$<Li#itControlLo?er −==

,ereG revised$ revisedn are t,e revised values of t,e Process $veragePro&ortion NonJconforming and t,e $verage Sam&le Si'e res&ectivel).iv. > tend Control Aimits L nce t,e initial 8or ,istorical9 data are consistentl)

contained it,in t,e trial control limitsG e tend t,e limits to cover future&eriodsG i.e.G t,e Control Aimits become t,e &erating Control Aimits. T,efuture data ill be com&ared and evaluated against t,ese &erating ControlAimits.

T,e Control Aimits for t,e ongoing &eriods ma) be altered from t,ose develo&edduring t,e anal)sis &eriod b) c,anging t,e sam&le si'e.

Ste$ 2 . Inter$ret for Pro4ess C $ :ilit" .<,en all t,e Control issues ,ave been resolved 8s&ecial causes identifiedG anal)sed

corrected and recurrence &revented9G t,e Control C,art reflects t,e ProcessCa&abilit) as discussed belo L

i. Calculate Process Ca&abilit)• :or a & c,artG t,e Process Ca&abilit) 8in t,e same sense as t,e Ca&abilit)

develo&ed from #ariables Data9 is reflected b) t,e &rocess average nonJconformingG $ G calculated ,en all &oints are in control. T,is can bee &ressed as t e &ro&ortion conforming to s&ecification 81 J $ 9

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• nce t,e &rocess is demonstrating statistical controlG t,e remaining averagelevel of nonJconformities ill reflect t,e s)stemic causes of variation in t,e&rocess J t,e Process Ca&abilit). T,e t)&es of anal)sis &erformed indiagnosing t,e S&ecial Cause 8Control9 issuesG ,ic, focused ono&erationsG ill no longer be a&&ro&riate in diagnosing Common Causesaffecting t,e s)stem. nless action is directed to ards t,e s)stem itselfG noim&rovement in t,e &rocess ca&abilit) can be e &ected. AongJterm solutionsare necessar) to correct t,e sources of c,ronic nonJconformities.

• ProblemJsolving and Statisical Tec,ni7ues 8ParetoG Do>G Cause >ffectsGetc. J see $&&endi )' for more details9 can be ,el&ful. =o everG

understanding of t,e &roblems can be difficult ,en onl) attributes data areused. In generalG &roblemJsolving is aided b) going u&stream in t,e&rocess as far as &ossible to ard t,e source of sus&ected causes of

variationG and b) using variables data for anal)sis 8e.g.G in H and / c,arts9.iv. C,art $nal)se t,e /evised Process

• <,en s)stematic &rocess actions ,ave been ta enG t,eir effects s,ould bea&&arent in t,e control c,art t,e c,art becomes a a) of verif)ing t,eeffectiveness of t,e action.

• $s t,e &rocess c,ange is im&lementedG t,e control c,art s,ould bemonitored carefull). T,is c,ange &eriod can be disru&tive to o&erationsG&otentiall) causing ne control &roblems t,at could obscure t,e true effectof t,e s)stem c,ange.

• $fter an) s&ecial causes of variation t,at a&&ear during t,e c,ange &eriod,ave been identified and correctedG t,e &rocess ill be in statistical controlat a ne &rocess average. T,is ne average reflecting inJcontrol

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II0 >0 CONTROL C5ARTS 1OR ATTRIBUTES

The n$ Ch rt for Nu#:er Non 4onfor#in+

T,e n& c,art measures t,e number of nonJconforming 8discre&ant or soJcalleddefective9 items in an ins&ection lot. It is identical to t,e & c,art e ce&t t,at t,eactual number of nonJconforming itemsG rat,er t,an t,eir &ro&ortion of t,e sam&leG isrecorded. ;ot, & and n& c,arts are a&&ro&riate for t,e same basic situationsG it,t,e c,oice going to n& c,art if 8a9 t,e actual number of nonJconformities is moremeaningful or sim&ler to re&ort t,an t,e &ro&ortionG and 8b9 t,e sam&le si'e remainsconstant from &eriod to &eriod. T,e details of instructions for t,e n& c,art arevirtuall) identical to t,ose for t,e &Jc,art e ce&tions are noted belo .

Ste$ (0 G ther / tT,e guidelines for gat,ering data are similar to t,ose for & c,artsG e ce&t for

t,e follo ing &ointsL• T,e ins&ection sam&le si'es must be e7ual. T,e &eriod of subJgrou&ing

s,ould ma e sense in terms of &roduction intervals and feedbac s)stemsGand sam&les s,ould be large enoug, to allo several nonJconforming itemsto a&&ear in eac, subgrou&. /ecord t,e sam&le si'e on t,e form.

• /ecord and &lot t,e number nonJconforming in eac, subgrou& 8n&9.

Ste$ -0 C l4ul te Control Li#itsT,e guidelines are similar to t,ose for & c,artsG it, t,e follo ing c,angesL

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,ere n t,e subgrou& sam&le si'e.

Ste$ )0 Inter$ret for Pro4ess ControlT,e guidelines are similar to t,ose for & c,art

Ste$ 20 Inter$ret for Pro4ess C $ :ilit"T,e guidelines are similar to t,ose for & c,artG it, t,e follo inge ce&tionLT,e &rocess ca&abilit) is n&G t,e average number nonJconforming in afi ed sam&le si'e n. T,is could also be e &ressed as t,e averagenumber conformingG n81J&9 nJn&.

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SECTION II0 , CONTROL C5ARTS 1OR ATTRIBUTES

The 4 Ch rt ; for Nu#:er of Non 4onfor#ities<

T,e cJc,art measures t,e number of nonJconformities Qdefects in an ins&ection lot

8as o&&osed to t,e number of units found nonJconformingG as &lotted on an n&

c,art9. T,e c c,art re7uires a constant sam&le si'e or amount of material ins&ected.

In s,ort c c,arts are used for controlling t,e variations in t,e number of defects in as&ecific &ortion of t,e &o&ulation.

It is a&&lied in t o major t)&es of ins&ection situationsL

• <,ere t,e nonJconformities are scattered t,roug, a moreJorJlesscontinuous flo of &roduct 8e.g.G fla s in a bolt of vin)lG bubbles in glassG ors&ots of t,in insulation on ire9G and ,ere t,e average rate of nonJconformities can be e &ressed 8e.g.G fla s &er 1 s7uare meters of vin)l9.

• <,ere t,e nonJconformities from man) different &otential sources ma) befound in a single ins&ection unit 8e.g.G t,e riteJu&s at a de&artmental re&airstationG ,ere eac, individual ve,icle or com&onent could ,ave one or moreof a ide variet) of &otential nonJconformities9.

T,e follo ing are t,e ste&s in construction and a&&lication of a c c,artG ,ic, aresimilar to t,e basic a&&roac, described &reviousl) for & c,arts

Ste$ (0 G ther / t

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cccc +++= ....21

,ere c1G c2 " are t,e number of nonJconformities in eac, of t,e subgrou&s.

• Calculate t,e Control Aimits 8 CAc and ACAc9ccCA c 3+=

ccACAc 3−=

Ste$ )0 Inter$ret for Pro4ess Control

T,e inter&retation is similar to t,at for & c,art.

Ste$ 20 Inter$ret for Pro4ess C $ :ilit"

T,e &rocess ca&abilit) is c G t,e average number of nonJconformities in a sam&le offi ed si'e n. It is similar to t,at mentioned for & c,art it, t,e above e ce&tion 0

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SECTION II0(' . CONTROL C5ARTS 1OR ATTRIBUTES

The u Ch rt for Non 4onfor#ities $er unit

T,e u c,art measures t,e number of nonJconformities 8discre&ancies or soJcalleddefects9 &er ins&ection re&orting unit in subgrou&sG ,ic, can ,ave var)ing sam&lesi'es 8 or amounts of material ins&ected9. It is similar to t,e c c,art e ce&t t,at t,enumber of nonJconformities is e &ressed on a &er unit basis. ;ot, u and c c,arts area&&ro&riate for t,e same basic data situations ,o everG t,e u c,art ma) be used ift,e sam&le includes more t,an one unit 8to ma e t,e re&orting more meaningful9Gand it must be used if t,e sam&le si'e can var) from &eriod to &eriod.e.g. T,e number of customers visiting a teller counter ma) var) ,our to ,our andt,e) ma) ,ave different t)&es of com&laints regarding t,e service 7ualit) of t,eban .e.g. $ job s,o& t)&e foundr) ma) &roduce different si'es of castings and t,e numberand t)&e of defects on eac, casting ma) var) from job to job.T,e above t o e am&les &resent t,emselves as cases for a&&l)ing t,e u c,art.T,e details of instructions for t,e u c,art are similar to t,ose for t,e & c,art

e ce&tions are noted belo L

Ste$ (0 G ther / t

• Sam&les do not need to be constant from subgrou& to subgrou&G alt,oug,maintaining t,em it,in 2!M above or belo t,e average sim&lifies t,e calculationof control limits.

• /ecord and &lot t,e nonJconformities &er unit in eac, subgrou& 8u9L

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,ere c1Gc2" and n1G n2 .. are t,e number of nonJconformities and sam&lesi'e of eac, of t,e subgrou&s.

• Calculate t,e control limits 8 CA and ACA9

nu

uCA u 3+=

nu

uACAu 3−=

,ere n is t,e average sam&le si'e.

0emoL If an) individual subgrou& sam&le si'e is more t,an 2!M above or belo t,eaverage sam&le si'eG recalculate t,e &recise control limits as follo sL

nu

uACACA uu 3, ±=

,ere u is t,e &rocess average and n is t,e sam&le si'e 8number of ins&ectionre&orting units9 of t,e &articular subgrou&. C,ange t,e limits on t,e c,art and use ast,e basis for identif)ing s&ecial causes.

Note t,at an) use of variable control limits is cumbersome and &otentiall) confusing.It is muc, better ,erever &ossible to avoid t,is situation b) using constant

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C,art inter&retations amount to recognition of unusual &atterns it, &rocessno ledgeG e &erience and a&&reciation of &robabilit). T,e illustrated &atterns are

b) no means all t,at can occur. $n)t,ing t,at loo s unusual s,ould be investigatedit, reference to t,e logG even if onl) to confirm an occurrence ,as no &articular

causeG or t,at a mista e ,as been made in measurementG calculation or &lotting. T,e&atterns s,o n belo can also occur b) c,ance causesG it,out t,ere being as&ecial disturbance.Care ,as to be ta en to avoid over inter&retation of t,e data &atternsG as evenrandom &atterns 8i.e.G due to common causes9 can sometimes give an illusion ofnonJrandomness 8i.e.G &resence of s&ecial causes9.

Situ tion $&&earanceC,ance of

:alse $larm

Point8s9 e ceeding eit,er of t,e ControlAimits

L :la ed materialG bro en toolGo&erator errorG &o er failure

On4e in:out =2'

$lots

out of ) su44essi%e $lots in the A On4e in E S

A%e

LCL

UCL

C

A

B

C

B

A

Z O N E S

Z O N E S

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3 4onse4uti%e $lots outside the4entr l ( ) rd ;to$ nd or :otto#< L suall) t,e result of mi ed sam&lesfrom different &eo&le or different

On4e in:out )-'

$lots

+ successive &lots 8* intervals9 in a roabove or belo t,e average line8centre line9 L Process c,ange caused b) resetting

=0> $erthous nd

Situ tion $&&earanceC,ance of

:alse $larm

* successive &lots 8( intervals9consistentl) rising or falling L >7ui&ment Q tool ear

On4e in:out =-'

$lots

A%e

LCL

UCL

C

A

B

C

B

A

Z O N E S

Z O N E S

A%e

LCL

UCL

C

A

B

C

B

A

Z O N E S

Z O N E S

UCLA

B

Z O N E S

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A%e

LCL

UCL

C

A

B

C

B

A

Z O N E S

Z O N E S

$lternating ,ig, Q lo &lots L suall) denotes an e tremee am&le of mi ed sam&les fromdifferent &eo&le or different mac,inesGetc. and are caused b) unsuitablec,artsG &rocedures and Q or standards.

Situ tion

$&&earanceC,ance of

:alse $larm

T,e &atterns belo are e am&les of situations t,at are not necessaril) outJofJcontrol and could beentirel) &redictable. T,ese &atterns onl) illustrate situations ,ere more com&le &rocess controlis necessar).

T,is &attern is oving eans L /eflects action to control in,erentl)unstable &rocesses.

A%e

LCL

UCL

C

A

B

C

B

A

Z O

N E S

N E S

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/es4ri$tion P ttern A$$e r n4e

T,e table belo s,o s &atterns of c,aracteristics of s&ecial events

T,is &attern ma) come from t odifferent mac,ines or material lots.Note t,e run can be se&arated as t odata it, different means.

$l a)s ,ave se&arate c,arts for eac,or station.

0i ture

LCL

A%e

UCL

T,is &attern indicates t,at t,ere is adeterioration over time 8e.g.G tool earGc,emical saturation going lo G etc.9

Trend

LCL

A%e

UCL

T,is &attern is often observed in batc,&rocesses 8nested sources9. T,econtrol limits do not account for all t,evariation sources. T,e data source isfrom one &rocessG but t,e control limitsare too tig t

0ulti&levariationsources

A%e

UCL

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> am&les of nonJrandom &atterns could be obvious trendsG t,e overalls&read of t,e data &oints it,in control limitsG or relations,i& among values

it,in subgrou&s. ne test for t,e overall s&read of t,e subgrou& data &lotscan be as s,o n belo . :or a statisticall) inJcontrol &rocessG t,e &lot&ro&ortions s,ould satisf) t,e follo ing L

If substantiall) more t,an 2Q3 of t,e data &oints lie close to 8for 2! subgrou&sif over + M are in t,e middle t,ird of t,e control limit region9G c,ec to see

,et,erL• T,e control limits or &lot &oints ,ave been miscalculated or mis&lottedG or

ACA

$ve

CA

C

$

;

CJ

;J

$J

2 of 3&lots

1+ of 2

&lots

3 in 1 outside t,e lines

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Pro4ess C $ :ilit"

In &lanning t,e 7ualit) as&ects of manufactureG most im&ortant is advanceassurance t,at t,e &rocesses ill be able to ,old t,e tolerances. Process ca&abilit)conce&t &rovides a 7uantified &rediction of &rocess ade7uac).

Process is a uni7ue combination of mac,ineG toolsG met,odsG material and t,e&eo&le engaged in &roductionG ,ic, converts in&ut into out&ut.

Precise but Inaccurate Im&recise Inaccurate

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Process ca&abilit) is determined b) t,e total variation t,at comes from commoncauses E t,e minimum variation t,at can be ac,ieved after all s&ecial causes ,avebeen eliminated.T,us ca&abilit) re&resents t,e &erformance of t,e &rocess itselfG as demonstrated

,en t,e &rocess is o&erated in a state of statistical control.

Ca&abilit) is often t,oug,t of in terms of &ro&ortion of out&ut t,at ill be it,in&roduct s&ecification tolerance. Since a &rocess in statistical control can bedescribed b) a &redictable distributionG t,e &ro&ortion out of s&ecification can beestimated from t,is distribution.

$s long as t,e &rocess is in statistical controlG it ill continue to &roduce t,e same&ro&ortion of out of s&ecification limits.

In s,ortL t,e &rocess must first be broug,t into statistical control b) detecting andeliminating s&ecial causes of variation. T,en its &erformance is &redictableG and itsca&abilit) to meet customer e &ectations can be assessed. T,is is a basis forcontinuing im&rovement.

Inter$ret tion of Pro4ess C $ :ilit"

=aving determined t,at t,e &rocess is in statistical controlG t,e 7uestion still remains,et,er t,e &rocess is ca&ableR Does its out&ut meet customer needsR Ca&abilit)

reflects variation from common causes and management action on t,e s)stem isalmost al a)s re7uired for ca&abilit) im&rovement.

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σ *LSLUSL C $ =

T,e C& inde determines t,e central tendenc) of t,e &rocess. Not onl) t,e &rocess,as to be ca&able G but it ,as to meet t,e s&ecifications. It is given b) t,e follo ingformula

σ σ )LSLMe n

nd )

Me nUSL of Mini#u#9C $ =

/

) Me n s$e40li#itNe rer

9C $σ

=

T,e &rocess ca&abilit) information serves t,e follo ing &ur&oses

1. Predicting t,e e tent of variabilit) t,at &rocess ill e ,ibit. Suc, information ,el&sdesigners to set realistic s&ecification limits.

1. Selecting t,e most a&&ro&riate &rocessG from t,e e isting ones.

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:our e am&les of &rocess variabilit)

C& Total amount outsidelimits

T)&ical actions to be ta en

X1. ≥!. M =eav) &rocess controlG sortingre or

1. .3 M =eav) &rocess control G Ins&ection

1.33 %4 PP0 /educed ins&ection G selected useof control c,art

1.%3 1PP0 S&ot c,ec ing Gselected use ofcontrol c,arts

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T,e goal of a &rocess control s)stem is to ma e economicall) sound decisions aboutactions affecting t,e &rocess. T,is means balancing t,e ris s of ta ing action ,enaction is not necessar) 8overcontrol or t,e T)&e I error9 versus failing to ta e action

,en action is necessar) 8undercontrol or t,e T)&e II error9. T,ese ris s must be,andledG ,o everG in t,e conte t of t,e t o sources of variation &reviousl)mentioned J s&ecial causes and common causes.

$ &rocess is said to be o&erating in a state of statistical control ,en t,e onl) sourceof variation is common causes. ;ut a state of statistical control is not a natural statefor a manufacturing &rocess. It is instead an ac,ievementG arrived at b) eliminationGone b) oneG b) determined effortG of s&ecial causes of e cessive variation.T,e initial function of a &rocess control s)stemG t,enG is to &rovide a statistical signal

,en s&ecial causes of variation are &resentG and to avoid giving false signals ,ent,e) are not &resent. T,is ill enable a&&ro&riate action t,at can eliminate t,oses&ecial causes and &revent t,eir rea&&earance.Process ca&abilit) is determined b) t,e total variation t,at comes from commoncausesG i.e.G t,e minimum variation t,at can be ac,ieved after all s&ecial causes,ave been eliminated. T,us ca&abilit) re&resents t,e &erformance of t,e &rocessitselfG as demonstrated ,en t,e &rocess is being o&erated in a state of statistical

control. Ca&abilit) is often t,oug,t of in terms of t,e &ro&ortion of out&ut t,at ill beit,in &roduct s&ecification tolerances. Since a &rocess in statistical control can bedescribed b) a &redictable distributionG t,e &ro&ortion of outJofJs&ecification &artscan be estimated from t,is distribution. $s long as t,e &rocess remains in statisticalcontrolG it ill continue to &roduce t,e same &ro&ortion of outJofJs&ecification &arts.0anagement actions to reduce t,e variation from common causes are re7uired toim&rove t,e &rocessF abilit) to meet s&ecifications consistentl).In s,ortG t,e &rocess must first be broug,t into statistical control b) detecting and

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=o everG from t,e above e &lanation of C $G it is obvious t,at C $ can onl) &rovideinformation about 5 ,at t,e Process can do6 and not,ing about 5 ,at t,e &rocess isdoing6. See t,e follo ing illustrationsG

In all cases 8$9G 8;9G 8C9 t,e t,ere is no c,ange in t,e values of 5>ngineeringTolerance6 and t,e 5Process <idt,6. hus, the ( p value *ill be constant and samefor 0A1, 021 as *ell as 0(1 . =o everG it is obvious t,atG

• Case 8$9 L T,e &rocess ,as andered to ards t,e Ao er S&ecification Aimit8ASA9G as suc, a &ro&ortion of t,e out&ut ill be belo t,e ASAG ,ic, illnot satisf) t,e >ngineering S&ecifications.

• Case 8;9 L T,e &rocess is centered and t,e Process <idt, is e7ual to t,e>ngineering S&ecificationsG as suc, almost t,e entire &o&ulation 8++.(3M9 oft,e out&ut ill satisf) t,e >ngineering S&ecifications.

• Case 8C9 L T,e &rocess ,as andered to ards t,e &&er S&ecification Aimit8 SA9G as suc, a &ro&ortion of t,e out&ut ill be above t,e SAG ,ic, illnot satisf) t,e >ngineering S&ecifications.

2. nilateral TolerancesL :or unilateral tolerancesG onl) one S&ecification Aimit

LSL USL LSL USLLSL USL

En+ineerin+ Toler n4e

Pro4ess 7idth

En+ineerin+ Toler n4e

Pro4ess 7idth

En+ineerin+ Toler n4e

Pro4ess 7idth

;A< ;B< ;C<

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• +arger the 2etter L =ere onl) ASA ist,e least value t,at t,e &rocess&arameter is e &ected to assume.In suc, casesG

7idthPro4ess -(

LSL C$L

- μ=

σ )

LSL C $L

- μ=

Pro4ess Potenti l IndeF C $9 LC $9 is a measure of ,o ell an inJcontrol Processis generating a given c,aracteristicG com&ared to t,e S&ecification Aimits. ProcessCa&abilit) Inde G

• Is a com&arison of t,e actual distribution of t,e Production ut&ut to t,e

s,ortest distance of t,e centre 8mean9 of t,e distribution from t,eS&ecification Aimits.

• Includes in t,e assessment t,e Aocation of t,e Process 0ean it, res&ect tot,e >ngineering Target or t,e mid&oint of t,e >ngineering Tolerance.

• 0easures ,at an inJcontrol Process is doing• De&ends on t,e Process Aocation as ell as t,e Process S&read

T,e ste&s involved in calculating C $9 are as follo sG

LSL

Q LSL Q

)$LC =

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• Calculate Standard Deviation 8 σ9 ofC,aracteristic

Use -dR

N = or 24s

N =

T,e total Process <idt, in case of a Normal Distribution is * G ,ence,alf t,e &rocess idt, ill be )

• Calculate C $9 as t,e ratio oft,e distance or

to ,alf t,e Process

idt, i.e.G

2. nilateral Tolerances L $s described aboveG for unilateral tolerancesG onl) one

CSA J μ CSA J X

Z3

( ))

CSL H or

)

CSL 9C$

=

/ist0 of 4losest S0L0 fro# the Me n

( - Pro4ess ?idth

LSL USL

Hor

Z3

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• +arger the 2etter L =ere onl) ASA ist,e least value t,at t,e &rocess&aramater is e &ected to assume.In suc, casesG

) LSL

9C $ =

Aet6s loo at t,e Process Potential Inde G C $9 from t,e ot,er &ers&ective E ,o can itbe vie ed as an indicator for estimating t,e &ro&ortion of t,e &rocess out&ut be)onda &articular value of interest 8e.g.G S&ecification Aimit9. Aogicall)G ca&able &rocessesare t,ose t,at ,ave virtuall) all t,e out&ut it,in t,e s&ecified >ngineering Aimits.Consider t,e follo ing illustrationG

LSL

Q LSL Q

)$LC =

LSL USL LSL USLLSL USL LSL USL

;/<;C<;B<;A<

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interestof % luethe&Me nPro4ess:et?eendist n4e =

T,usG t,e distance of bot, t,e S&ecification Aimits 8 SA ASA9 from t,e Process0ean 8 9 ill res&ectivel) beG

N

H B USL C

USL =

NLSL B H

LSL =

T,is leads us to loo at t,e Process Potential Inde G C $9 differentl). Com&aring t,eB SA B ASA to identif) t,e minimum of t,e t oG i.e.G B min leads us to ,aveG

)

9C #in$ =

B values can be used it, a Table of t,e Standard Normal Distribution 8$&&endi)'9 to estimate t,e &ro&ortion of out&ut 8an a&&ro imate valueG based on t,e

assum&tions t,at t,e &rocess is in statistical control and is normall) distributed9 illbe be)ond an) S&ecification Aimit or a value of interest. :ollo ing is t,e met,odem&lo)edG

• :or nilateral Tolerances L Aocate t,e value of B on t,e column on t,e left edgeof t e table 8t e units tent s digits9 T e undredt s digits are in t e

Kor μ

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•( (

or 3ASAJK

3

KJSA of min. C&

=

σσ

=

3B

3

B of min. ASASA

(0))T,usG ,en min # 5 G( p # $.55

1ere the process has been centered and the %ariability has been impro%ed sothat the nearest Specification Limit (in this case both) is 7 σ a*ay from the

rocess Mean i.e. the rocess output is occupying only 895 of the $olerance6and.

T,usG it is eas) to understand t,e significance of relations,i& of ProcessCa&abilit) Inde 8C & 9 value in terms of B.

If t,e Process 0ean is centered 8or controlled at desired location9G AongJtermProcess Im&rovement involves reduction in t,e variabilit) 8s&read9. T,e focusno could be s,ifted to anot,er Ca&abilit) Criterion of K ±3σ 8Bmin ≥ 3 C& 1. 9G or K ±4σ 8C& 1.339 """"G instead of a target value of ProcessCa&abilit) Inde . In ot,er ordsG s,ifting t,e focus from 5 Monitoring 6 to5Controlling 6G from 5Effects 6 t o 5Causes 6G from 5#utputs 6 t o 5Inputs 6G from5Symptoms 6 to 5 roblems 6G i.e. to focus im&rove B and C & ill automaticall) getmonitored U Y f;H<VT,usG for an im&rovement in C & valueG if t,e Ca&abilit) Criterion in terms of B is

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!ample :: $ !3. $he tolerances are bilateral and process target is desired at the centre of the

Specification Limits;. $he current rocess %erage < 4.82=2. $he >pper Specification Limit (>SL) < 4.?44 7. $he Lo*er Specification Limit (LSL) < 4.944 9. $he current rocess Standard :e%iation 4.48;9 @. $he Capability Criterion < A ± 7σ

" LBSIS !3. From the abo%e it is ob%ious that the process performance is not satisfactory

and

• ( );.;2

4.82= (4.?44 A (>SL C

>SL ===!"2#.!σ

i.e. the >SL is ;.;2 σ a*ay from

the current process a%erage. $hus about 3.;?5 of the output is beyond the>SL.

• ( ) 2.;= 4.944 (4.82=LSL( A C LSL ===

!"2#.!σ i.e. the LSL is 2.;= σ

a*ay from the current process a%erage. $hus about 4.49;5 of the output isbeyond the LSL.

• $otal output beyond Specification Limits (and may probably need sorting ifno capability impro%ement is done) *ill be 3.27;5 (327;4 M)

;. "o* e%en if the process is shifted to the centre ( A < 4.844)

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( )7

4.844 (4.?44=

4.4944 =

&( )

7

4.944 (4.844=

4.4944 =

$hus 4.4944 ne* =σ

9. $his means that actions must be ta+en to reduce the rocess Standard:e%iation from the current %alue of 4.48;9 to 4.4944

• 235 344 4.48;9

4.4944 (4.48;9 344

(

old

ne* old =

=

σ

σ σ

"ote ! In some cases a short term alternati%e could also be thought in terms ofre loo+ing into the tolerances for an increase. $he t*o possibilities could be

• If the process is not to be changed the ne* specifications *ould be

( ).2+ .(3* . (2!4 K ±=×±

$he re%ised >SL < 3.4;= & re%ised LSL < 4.77= • If the process has been changed (and confirmed by the control charts) so

that ne* A < 4.844

( ) ( ) .2+.( . (2!4 .( . (2!4 K ±=×±=×±

$he re%ised >SL < 4.??4 & re%ised LSL < 4.734

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T)&ical $reas under t,e Normal Curve

It ,as been observed t,atG over timeG an) inJcontrol &rocess is li el) to s,ift b)1.! σ . :rom t,e follo ing illustrationG it is obvious t,at at 3 σ ualit) level 8 it,outc,ange ould give us about 2( &er million defectives. =o everG if a t)&icalc,ange in average is allo edG t,e defectives jum& to %%* 3 &&m. Consider aProcess at % σ ualit) levelG i.e.G case 8;9G even after factoring for t,e t)&ical

f i ti G t d f ti ld b l) t 3 4 && 8f 2 && 9

*>0-*

,302*

,,0=)

,,0,,)=

,,0,,,,2)

,,0,,,,,,>

( σ - σ ) σ 2σ 3 σ * σ* σ 3 σ 2σ ) σ - σ ( σ '

∞ ∞

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SECTION III0) PROCESS CONTROL & PROCESS CAPABILITY

Preli#in r" Pro4ess C $ :ilit"

T,e ultimate criterion of &rocess management is continuing im&rovement in t,eabilit) of &roducts to e ceed customer e &ectations using efficient &rocesses. T ointernal criteria must su&&ort t,isL &rocess stabilit) and &rocess ca&abilit). Stabilit) isa &rere7uisite and must be attained first. T,e &rocess must be broug,t into astatistical control 8 all s&ecial causes removed9. Process ca&abilit) can t,en beassessed to determine t,e e tent of common cause of variation.

T,ere are t o as&ects of &rocess ca&abilit)L• Preliminar) ca&abilit)• ngoing ca&abilit)

Preli#in r" Pro4ess C $ :ilit">ffective 7ualit) &lanning e ecution s,ould )ield &rocesses ,ic, are bot, stable

ca&able. T,e e tent to ,ic, t,ese goals ,ave been ac,ieved is establis,ed for

bot, &rocess stabilit) and ca&abilit) b) a ca&abilit) stud) carried out before full&roduction begins. T,is stud) called as &reliminar) &rocess ca&abilit) stud)G&rovides a &reliminar) indication of t,e abilit) of a &rocess to &roduce &arts ,ic,are constantl) it,in s&ecifications.

f necessit)G it ill be of limited duration and ill not full) re&resent ongoing&roduction conditions. =o ever it does offer timel) information on &rocess&erformance and ,ere o&&ortunities for im&rovement lie. T,is stud) can usefull)ta e &lace at man) stages. :or e am&le

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If t,ese indicate t,at t,e &rocess is ca&ableG a&&rove t,e &rocess and see furt,ero&&ortunities for im&rovement. If t,e &rocess is not ca&able underta e efforts toim&rove t,e &rocess so t,at ca&abilit) is ac,ieved.

Condu4tin+ $reli#in r" $ro4ess 4 $ :ilit" stud"

bjectives• To obtain earl) information on ne or revised &rocess• To develo& &reliminar) control c,arts• To evaluate &rocess stabilit)• To assess acce&tabilit) for ongoing &roduction

Sco&e• $&&licable to eac, critical significant c,aracteristics ,ere C& value is

un no n• 0a) be conducted at several &oints in evolution of ne &rocess

-at,er data• $t least 2 rational subgrou&s of 3 to ! measurements in eac,• If data is more limited start control c,art an),o

>stablis, stabilit)• Plot on control c,art• >nsure stabilit) &rior to ca&abilit) anal)sis

Determine &erformance

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C l4ul te Indi4esProcess &otential and ca&abilit) indices are used to &rovide a standardi'ed measurefor e &ressing t,e &erformance of a &rocess in relation to s&ecified re7uirement.

:our sets of indices involved areL

IndicesPotential Ca&abilit)

8measures &recision measures &recisiononl)9 and setting

Stud)

Preliminar) P& P&

ngoing C& C&

Im&ortant noteL T,ese indices ma) be calculated onl) ,en a control c,art indicatest,at stabilit) ,as been ac,ieved i.e. &rocess is in statistical control 0

C& Process &otential inde 0C& is an inde G ,ic, is ratio of t,e s&ecified tolerancerange to t,e si standard deviation &rocess s&read 8or e7uivalent nonJnormaldistribution it,out regard to t,e location of data.9

C& &rocess ca&abilit) inde . C& is an inde G ,ic, considers bot, &rocesss&read and t,e location 8setting9 of data in relation to t,e s&ecification limits 0

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Protot"$e $ rtsPISTL It is an abbreviation for &er cent of Ins&ection Points ,ic, Satisf) Tolerance.It is an inde G ,ic, describes t,e e tent to ,ic, t,e first run of &arts 8&rior to

necessar) re or 9 meet t,e &rint re7uirements. T,e se7uence of events for t,ePIST inde calculations follo sL

1. Si &rotot)&e &arts are subjected to com&lete la)out c,ec . $ll dimensions andnotes on t,e dra ing are c,ec ed.

2. PIST inde is calculated. PIST inde is calculated as t,e ratio of number ofconforming ins&ection &oints to total number of ins&ection &oints into 1 .

(''F# dets#e sure#enof nu#:er

o9reth tts#e sure#enof Nu#:er PIST =

=ence PIST is re&orted as &ercentage.

PIPC L It is an abbreviation for Percent of IndicesG ,ic, are &rocess ca&able.PIPC is t,e &ercentage of critical or significant c,aracteristics it, C& C& indicesgreater t,an or e7ual to 1.33 in t,e &rotot)&e and &roduction &,ases.8alternativel)G P& and P& indices ≥ 1.%(.PIST is &art s&ecific and relates to all dimensions. PIPC is &rocess oriented andrelates to critical and significant c,aracteristics. T,e follo ing figure gives a clearidea of bot,.

PIST PIPC /eri% tion

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:ollo ing bar c,art illustrates t,e PIST ualit) ;enc,mar s

1J>PJ >valuation &rotot)&e 2J#PJ #erification &rotot)&e 3J:;J :unctional ;uild 4J /egular su&&l)

PIST > am&lePIST e7uals t,e number of conforming ins&ection &oints divided total number ofins&ection &oints multi&lied b) 1 :or t is e am&leG data from si &arts is s o n

PIST !UALITY BENC5MAR6S

(*!

1 1

2

4

%

*

1

12

1 2 3 4

P e r 4 e n

t + e

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>P #P are &rotot)&e &,ases and :;G P / are &roduction &,ases.T,e c,art s,o n above illustrates t,e PIPC goals. T,e ,eig,t of bars re&resents t,e&ercent conformance relative to goal. T,e PIPC c,art s,o s a goal of 1. for C&

PIPC !UALITY BENC5MAR6S

.2

.4

.%

.*1

1.2

1.4

>P #P :; P /

Protot"$es

P r o 4 e s s

C $ :

i l i t "

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4.$ddress ]]]]]]]]]]]]]]]]]]]]] ]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

#e,icle ;uild8Circle one9L >P #P :; t,er 8Define9 C,aracteristics T)&eC.C. E 0 0 critical S.C. 0 0 Significant c,aracteristics c,aracteristics

Number of critical andQor significant c,aracteristicsL1 T)&e C& C&2 T)&e C& C&3 T)&e C& C&4 T)&e C& C&! T)&e C& C&% T)&e C& C&( T)&e C& C&* T)&e C& C&+ T)&e C& C&

1 T)&e C& C&

PIST (*Q+ 1 *%.( M PIPC c& 23Q2( 1 *!.2M PIPC c& 2 Q2( 1 (4M

Comments

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-enerall)G t,ere is a difference bet een mac,ines and &rocesses for statisticalanal)sis. 0ac,ines are regarded as discrete o&erationsG for e am&le a drill or alat,e. Processes are considered to contain a combination of &eo&leG e7ui&mentG

materialsG met,ods and environment.

Ca&abilit) is measured relating t,e actual &erformance of a mac,ine or &rocess toits s&ecified &erformance.

:or variables 8measured data9 t,e ca&abilit) of a mac,ine is a measure of t,e s,ortJterm influences on &roduct variabilit) emanating mainl) from t,e mac,ine. T,eminimum re7uirement is t,at ± 4 standard deviations are contained it,in t,es&ecification limits. T,is means t,at ++.++4M are it,in tolerance. T,is morestringent re7uirement is necessar) to indicate t,at t,e goal of ++.(3M 8 ± 3standard deviations9 &rocess ca&abilit) could be ac,ieved over t,e long term. T,is isstill no guarantee of continued conformance to s&ecification. To ma imiseconformance to s&ecificationG continuous im&rovement of ca&abilit) must be&ursued.

In generalG &rocess ca&abilit) im&rovement can be ac,ieved b)L• >nsuring t,at all s&ecial causes of variation ,ave been eliminated.• >nsuring t,at t,e &rocess means is targeted to ard t,e s&ecified nominal

value.• /eduction of common cause variation.

:or attribute dataG ca&abilit) is a measure of t,e conformit) rate of t,e mac,ine&

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$B( 4 $ :ilit" =

:or c and u c,arts ca&abilit) cannot be e &ressed in t,e same manner. C and u are

usedG res&ectivel)G as measures of &rocess &erformance.

Note. $n) ca&abilit) anal)sis tec,ni7ueG no matter ,o &recise it a&&earsG can giveonl) a&&ro imate results. T,is ,a&&ens because 819 t,ere is al a)s some sam&lingvariationG 829 no &rocess is ever full) in statistical controlG and 839 no out&ut e actl)follo s t,e Normal distribution 8or an) ot,er standard distribution9. :inal resultss,ould al a)s be used it, caution and inter&reted conservativel).

/0 Introdu4tion to C $ :ilit" Studies

0ac,ine ca&abilit) studies are means of assessing &rocess &otential over a s,ort&eriod of time from a relativel) small amount of sam&le data. $ s,ortJterm stud) oft,is t)&e ill give an estimate of t,e variation due onl) to t,e mac,ine itself. It ill notreflect ot,er &rocess influencesG ,ic, var) over time. :or t,is reason t,ere7uirement is for ^ 4 standard deviations to lie it,in s&ecification. T,e) aret)&icall) used as &art of a large investigation of t,e &rocess or as a means ofassessing t,e &erformance of a ne mac,ine or &rocess.

ne met,od of assessing mac,ine ca&abilit) is a gra&,ical straig,t lineO met,odusing t,e &ro&erties of &robabilit) &a&er. T,e Normal cumulative M &robabilit) scales,o n in :igure readil) transforms t,e bell s,a&ed Normal distribution into a straig,tline. It &ermits t,e visuali'ation of a Normal distribution in terms of a straig,tJlinereference standard.It is desirable to or in terms of a straig,tJline reference standard for a number of

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E0 M 4hine C $ :ilit" Stud" EF #$le

Ste$ (0 Pre stud" Re uire#ents

• Determine t,e number of &arts re7uired for t,e stud)G usuall) ! . If t,e &attern ofvariation is e &ected to be nonJNormalG a sam&le si'e of at least ! is mandator)Gand even greater is desirable.

• >nsure t,at all materials ,ave been c,ec ed and a&&roved for conformance tos&ecifications.

• >nsure t,at t,e measuringQtesting e7ui&ment ,as measuring &recision of at least1Q1 of t,e s&ecified tolerance.

• Ta e ste&s to ensure an uninterru&ted runG under normal &roduction conditionsGit, t,e mac,ine set at nominal.• In a multi&le fi ture setJu&G treat eac, station as a se&arate mac,ine for mac,ine

ca&abilit) &ur&oses.

Ste$ -0 Colle4t nd Stru4ture / t

• 0easure t,e sam&le out&ut of t,e mac,ine and recordG in se7uenceG on&robabilit) or s,eet.

• 0ar off t,e measurement scale to embrace t,e full range of readings. $guide to t,e number of class intervals in t,e scale can be derived from t,es7uare root of t,e number of readings in t,e sam&le 8 9. >.g. for a sam&leof ! G ( class intervals ould be a good starting &ointG ,ilst 1 classintervals could be more a&&ro&riate for a sam&le of 1 . Normall)G t,escale ill be ta en at t,e level of &recision at ,ic, t,e measurements are

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located on t,e gra&, along t,e ste&&ed line. 8NoteL ,en reading bac tot,e scaleG ignore t,e ste& u&9.

Colle4tin+ & stru4turin+ d t

C,aracteristic Dimension &eration SDPart Number Name

#alue #alue #alue #alue #alue #alue #alue #alue #alue #alue

,3 >3

>3 ,'

,3 ,' et40

('3 ,3

,' (''

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Ste$ )0 Constru4t Pro: :ilit" Plot

• :ollo ing t,e arro s across from t,e leftJ,and side to t,e rig,t ,and side oft,e formG &lot a &oint on t,e interce&t of t,e u&&er class interval and t,e&oint on t,e lo er scale coinciding it, t,e cumulative fre7uenc)&ercentage.

• /e&eat t,is &rocedure for all t,e &oints in t,e cumulative fre7uenc)&ercentage columnG e ce&t for t,e 1 M value 8for ,ic, t,ere is no &ointon t,e &robabilit) scale9. NoteL Aoss of data can be avoided at t,is &oint b)&lotting t,e average of t,e last t o &ercentages at t,e final interval mid&oint.

• If t,e &lotted &oints lie in a reasonabl) straig,t lineG dra a 5best fit6 straig,tline t,roug, t,em. > tend t,is line until it meets t,e vertical lines at t,ee tremes of t,e &ercentage scales 8^ 4s lines9.

• If a close fit is not &ossible t,is indicates t,at t,e data does not conform tot,e Normal distribution ,ic, ,as been assumed. :ollo ing &ages illdiscuss t,e a)s of ,andling t,ese situations.

Ste$ 20 Inter$ret for Confor# n4e to S$e4ifi4 tion

• If t,e e tension of t,e best fit line to t,e vertical lines at t,e e tremes of t,e&ercentage scale does not cross eit,er of t,e s&ecification limit linesG t,e mac,inecan be deemed to be conforming 8i.e.G ++.++4M of t,e &roduct ill fall it,ins&ecification9.

If t b tJfit li d it & ifi ti li it li G t i i t

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10 Non Nor# l / t

1. /easons for NonJNormalit)T,e &receding discussions ere based on a Normal distribution. <,ile t,is is t)&icalfor man) o&erationsG t,ere are situations in ,ic, t,e data are not Normall)distributed. T,ese include t,e follo ingL• T,e underl)ing distribution fits a standard statistical model ot,er t,an Normal.

:or e am&leL• :latnessG outJofJroundness or ot,er c,aracteristicsG ,ic, are bounded b) 'ero

oftenG ,as a s e &attern.• T,e basic underl)ing distribution is com&le and does not fit an) standard model.

:or e am&leL• Constraints it,in t,e mac,ineG suc, as internal sto&s or selfJadjusting ill

almost al a)s e ,ibit a nonJNormal &attern.• ut&ut ,ic, is t,e result of several &rocess ste&s or ,ic, is t,e combined

out&ut from several similar mac,inesG ma) not be Normall) distributedG even ifout&ut of t,e individual mac,ine is normall) distributed.

• 0ovement of meanG resultingG for instanceG from tool ear ma) cause nonJNormalit).

• >ven t,oug, t,e basic distribution of t,e mac,ine6s out&ut is NormalGs&ecial causes of variation ma) be &resent in t,e &rocess causingnonJNormal &atterns. :or e am&leL

• $ damaged tool or s arf in t,e locating device.

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Se4tion III0 3 E% lu tion of $ro4ess 4 $ :ilit" for Ch r 4teristi4s th t follo?S9e? /istri:ution

C& and C& formulae ,old good onl) ,en t,e distribution of t,e c,aracteristicfollo s normal distribution. T,is is because % measures &rocess ca&abilit) underassum&tion t,at t,e distribution of c,aracteristic follo s normal distribution. If t,ec,aracteristic is normall) distributedG &ercentage of items outside t,e s&ecificationsis al a)s .2( MG ,en C& Jis 1.

T,ere are some c,aracteristicsG ,ic, do not follo t,e normal distribution. T,ee am&les are some of t,e geometrical tolerances li e ovalit)G eccentricit) etc.-enerall) t,e) follo s e ed distribution. :or a s e ed distribution t,e &ercentoutside limits is more t,an .2(M even if t,e C& value is 1. =ence for a s e eddistribution t,e follo ing formula is used for calculation of &rocess ca&abilit)

k $$ C HC λ λ =

<,ere C& ,as conventional meaning and is a correction factor ,ic, is given b)t,e follo ing e7uationL

( ))$

)$9-)

6FC

,6FC(>;6

λ

λ 3

3) −++=

6 ) ≠ '

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:or a data let us assume t,at ma imum limit for out of roundness . !

. 1( G . 1( and @3 1.32

T,en C& 8 . ! E . 1(9Q 3 W . 1(

1.1

;) substituting t,e values of C& and @3 in t,e formula

.( !

C& C& 1.1 .( ! .(*

=ence actual &rocess ca&abilit) is .(* and not 1.1 calculated b) t,e normalformula.

It is im&ortant to note ,ere t,at C& overestimates t,e value if t,e &rocess does notfollo t,e normal distribution.

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Se4tion IV0 Ad% n4ed St tisti4 l Tools

(0 Correl tion & Re+ression

Correlation concerns t,e relations,i& bet een t o variables. /egression is t,e useof relations,i& to &redict t,e outcome of a &rocess. T,ere are t o a)s of describingt,e relations,i& bet een t o variables E gra&,ical and mat,ematical. -ra&,icaldescri&tion of correlation is scatter diagram.

(0 Correl tion

0at,ematical relations,i& bet een t o variables is described b) correlation.In &rocess control one aims to control t,e c,aracteristics of t,e out&ut of t,e &rocessb) controlling a &rocess &arameter. ne succeeds in t,is if t,e &arameters arecorrectl) c,osen.T,e c,oice is usuall) based on judgment and no ledge of t,e concernedtec,nolog). ne assumes correlation bet een a variable &roduct c,aracteristic anda variable &rocess &arameter. T,ere is a need to test if t,e assum&tion is correct&articularl) ,en one gets nonJconforming out&ut in s&ite of t,e &rocess a&&earingto be in control.T,is is ,) one needs to stud) t,e correlation bet een t,e t o variables.

Correlation Coefficient

T,e 7uantitative measure of correlation bet een t o variablesG t,e correlationcoefficientG is re&resented b) t,e s)mbol r.

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T,e total range of t,e value of r is E1 to 1. $ score of E1 indicates a ver) strongnegative relation meaning t,at if goes u& ) goes do n. $ score of 1 indicates astrong &ositive relations,i& ,ere and ) go u& or do n toget,er. $ score of

indicates absence of relations,i&. #alues closer to indicate mild relations,i& andt,ose closer to 1 indicate strong relations,i&.

T,e follo ing figures indicate gra&,icall) t,e inter&retation of t,e values.

r ( r '03 r '

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Re+ression

<,ile calculating t,e coefficient of correlationG also called regression coefficient8r9G

t,e data as fitted into a best fitting straig,t line. T,e formula t,at describes t,isstraig,t line ,as some constants. T,ese constants ,el& one &redict t,e value of a&roduct c,aracteristic for a given value of t,e &rocess &arameter. T,ese are t omajor uses of regression.

Aet us consider t,e e7uation of a straig,t line.

H mK c

In t,is K H are variables and m c are constants.

:or t,e sa e of convenience let us c,ange t,e nomenclature of constants to

H aK b

T,e formula for calculating a is

( )∑∑=

2HH

""HH

/

YFH

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Process &timi'ation

Process &timi'ation is reverse of &rediction. Aet us consider t,e e7uation

H aK b

HJb aK

r K 8HJb9 Q a

:or a desired value of &roduct c,aracteristicG e can select t,e &rocess &arameter.

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IV0 - 0 Ad% n4ed St tisti4 l Tools @ t @ Test

T,is is a&&lied ,en one ants to c,ec if t,e difference bet een t o &rocessaverages is significant or not. T,e t o averages to be com&ared ma) eit,er beeit,er be &rocess average of a modified &rocess and t,e average of traditional&rocess or averages of t o alternative &rocesses.5 t 5 test is a&&licable for variable data as ell as attribute data.T,is gives rise to four &ossible combinations

a. Traditional #s. 0odified &rocess J #ariable data

b. Com&arison of t o &rocesses J #ariable datac. Traditional #s. 0odified &rocess J $ttribute datad. Com&arison of t o &rocesses J $ttribute data

$s t,e a&&lication formula in eac, case is differentG eac, needs to be dealtse&aratel).

0 Tr dition l Vs0 Modified $ro4ess

T,e &rocess for a&&l)ing 5 t test to c,ec if t,e &rocess average in case of amodified &rocess ,as s,ifted significantl) from t,e traditional &rocess consists of t,efollo ing ste&sL

1. Collect data about t,e out&ut of t,e modified &rocess for t,e variable &arameterto be com&ared.

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Kb are individual values from &rocess ;

n a are number of sam&les from &rocess $ and

n b are number of sam&les from &rocess ;

3. Calculate t,e combined standard deviation of t,e t o &rocesses b) t,e formula

( ) ( )-nn

HHHH s

:

::: −+

+−= ∑∑

2a

,ere S ab is t,e combined standard deviation of t o &rocesses.

20 Calculate t,e value of 5 t 5 b) using t,e formula

:

:

:

nns

HH t

11 +=

!. Select t,e desired confidence level and find from t,e table t,e critical value for tfor 8 na nb J 29 degrees of freedom

*0 If calculated value of 5 t6 8ignoring t,e negative sign9 is larger t,an t,e criticalvalues in t,e tableG t,e ,)&ot,esis t,at t,e mean for t,e t o &rocesses is t,esame is rejected. If calculated value is smallerG t,e ,)&ot,esis is acce&ted.

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3. Calculate t,e value of 5 t 6 using t,e formula

<,ere &t is t,e value of t,e fraction defective mean for t,e traditional &rocess.

4. Com&are t,e calculated value of 6 t 5 for t,e selected confidence limit from t,etable and dra conclusions as earlier.

d0 Co#$ rison of t?o $ro4esses Attri:ute d t

T,e &rocedure for a&&l)ing t,e 5 t6 test for fraction defectives for t o alternative&rocesses is similar to t,e corres&onding &rocess for variable data. T,e ste&s of t,e&rocedure areL

1. Collect fraction defectives data for t,e t o &rocesses.2. Calculate values for mean fraction defective for t,e t o &rocesses.

3. Calculate6 t6 using t,e formula

n p p

t )1( −

= t$$

:$$t

−=

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IV0 ) Ad% n4ed St tisti4 l Tools Chi S u re Test

:or testing if t,e standard deviation of a &ro&osed ne &rocess is significantl) ,ig,er

t,an t,at of a traditional &rocessG K_ 8C,i S7uare9 test is a&&lied. It can also bea&&lied to test if t,e &ro&osed ne &rocess ill be able to &erform as e &ected b)t,e tolerance limits of t,e &arameter. T,e ste&s in t,e &rocedure for t,e a&&licationof t,e test are as follo sL

1. Collect data using t,e &ro&osed ne &rocess.

2. Calculate standard deviation from t,e data using t,e formula

3. Calculate t,e value of K_ using t,e formula

$re t,e standard deviations of t,e traditional and &ro&osed&rocesses res&ectivel).

( )1

2

−= ∑

n p

FF σ

( )t

(n

σ

σ 22 p X =

nd $t σ σ where

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:or com&aring t,e &recision of t o alternate &rocessesG : test can be used.T,e sam&le si'e or t,e degrees of freedom for bot, t,e &rocesses enter intoconsideration and ,enceG as e ,ad seen for t,e testG t,e formula for : test differs

from t,at of C,iJS7uare test. =ere even t,e table is different as bot, degrees offreedom ,ave to be considered for arriving at t,e critical value of :. T,e ste&s fora&&l)ing t,e : test are as follo sL

1. Collect data for t,e &arameter of interest for t,e out&ut of bot, t,e &rocesses andcalculate t,e standard deviation for bot, t,e &rocesses.

2. Calculate t,e value of : using t,e formula

T,is ensures t,at t,e value of : ill never be less t,an 1.3. Com&are t,e value of : calculated in ste& 2 it, t,e critical value of : from table

for +! M confidence level at a&&licable degrees of freedom for t o &rocesses.

4. If calculated value of : is ,ig,er t,an t,e value of 5:6 in t,e table fora&&ro&riate degrees of freedomG t,ere is sufficient evidence to reject t,e)&ot esis t at t e variance of t e t o &rocesses is t e same

one0lo?er theis

nd% luesde%i tionst nd rdt?otheof hi+her theis

t

h

σ

σ

σ

σ

where

F t

h

2

2

=

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A:str 4t

Statistical &rocess control 8SPC9 is a &o erful tec,ni7ue for im&roving &rocess7ualit) b) s)stematicall) eliminating s&ecial or assignable causes of variation. SPCis not a sim&le and automatic tas . T,e successful a&&lication of SPC re7uiresman) s ills suc, as engineeringG managementG statisticalG team or and &lanning.

$lt,oug, t,e use of control c,arts is a ver) im&ortant as&ect of SPCG it is b) nomeans t,e onl) one for t,e im&lementation of SPC in organi'ations. 0an)engineers toda) are e &osed to onl) control c,arts 8i.e. different t)&es9 and not t,eot,er im&ortant ingredients for t,e successful im&lementation of SPC. T,is article

ill briefl) discuss t,e ten e) ingredients t,at are needed for t,e effectivea&&lication of SPC in an) rgani'ation.

Introdu4tion

Statistical &rocess control 8SPC9 ,as been idel) acce&ted among 7ualit)&ractitioners as an aid for monitoringG managingG anal)'ing and im&roving &rocess&erformance b) eliminating s&ecial causes of variation. T,e use of ,ard SPC data&ermits a scientificG dataJ based management st)le in ,ic, decisions are madebased on factsG rat,er t,an guess or G and better &roducts can be &roduced it,less scra& and re or 8:ineG 1++(9. T,e &,iloso&,) under&inning t,e mec,anism ofSPC is t,at t,e &rocess out&ut can be broug,t into a state of statistical control b)means of engineering and management action. SPC is a re&lacement of t,etraditional a&&roac, to 7ualit) control 8sometimes referred to as ins&ection based7ualit) control9G ,ic, is rat,er e &ensiveG inefficientG unreliable and &rovides no

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&Jc,art9. -ood &ractice c,ooses variable measurements over attributeG as variabledata &rovides more information.

>0 Or+ niD tion l 4ultur l 4h n+e $ successful introduction of SPC ma) involve t,e cultural c,ange in t,e entireor ing environment. &eratorsG ,o no t,eir &rocesses better t,an an)bod)G

s,ould be em&o ered to ta e o ners,i& of t,e or and of t,e &rocesses t,e) dealit, ever)da). /ecognition and being a &art of t,e com&an) for em&lo)ees is

crucial. If t,e o&erator cannot tac le a s&ecific &roblemG t,en t,e &rocess actionteam is res&onsible to tac le it it, t,e su&&ort of steering committee andmanagement.

,0 Use of $ilot $ro e4tIt is not a good idea to a&&l) SPC in all de&artments at t,e same time. It ould bebetter if t,e steering committee could come u& it, a &ilot &roject to ac7uire ana&&reciation of t,e use of SPC as a &o erful &roblemJsolving tool. T,e mostim&ortant t,ing ,ere is to gain t,e attention of to& management and get ever)oneinterested in t,e area. nce SPC ,as been used successfull) in one &rocessG it ist,en easier to e tend its use to ot,er areas it,in t,e same de&artment and evenot,er de&artments. $ t)&ical &ilot stud) can generall) ta e from t,ree mont,s tomore t,an a )ear.

('0Co#$uters nd SPC soft? re $ 49 +esT,e use of com&uters ,as received muc, attention recentl) and ill be e tremel)useful in t,e near future. tili'ation of SPC soft are &ac ages 8e.g. 0INIT$;GST$TISTIC$G ST$T-/$P=ICSG S C P$C@G etc.9 ill assist t,e users in &lotting t,e

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7or9ed eF #$le 5isto+r #

T,e Telecom Com&an) anted to stud) t,e director) in7uir) ,andling times. T,e

data in a table is given in number of second re7uired to ans er an in7uir). Dra a,istogram and inter&ret it.

4% 3( (2 *1 !4 !% % %4 4%42 !2 + %( (2 34 *+ !* 3%!2 !! (% (3 * ( 2% !* 4%44 !2 %4 ( %4 ! 32 !2 3%!2 44 * !4 (1 + 21 ! 32!* !2 *% %% (% !* +( % 4%!% (1 %1 % ! * * %* 4%!! %2 !% 44 42 % (2 1+ (%! ( (! !2 4% 4% %% %4 *%1 2 3 %% 44 !% !* %% %* %+!! !2 3* 3! 42 4% +1 * %2+2 3* %2 44 4! 44 (* %1 (*

Ste$ ( 0

Count t,e number of observations 1 *

Ste$ -0

Decide number of intervals 5 S7uare root of 1 * 1 3+

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Ste$ 20

Tabulate fre7uenc) distribution

-rou& Seconds 0id &oint -rou& fre7uenc)1 1 E 1+.+ 1! 12 2 J2+.+ 2! 23 3 J3+.+ 3! 14 4 J4+.+ 4! 1*! ! J!+.+ !! 2%% % J%+.+ %! 21( ( J(+.+ (! 1!* * J* .+ *! ++ + J++.+ +! !1 1 J1 +.+ 1 ! 1 Ste$ 30

Dra ,istogram. T,is can be done in eit,er com&uter e cel standard &ac age8> ce&t t,e curve9 or on gra&, &a&er. Ta e time in seconds on KJa isG and grou&fre7uenc) on HJa is

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=istogram

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A$$endiF C Coeffi4ients for % ri :les 4h rts ; & R 9

Sub -rou&si'e Control Aimit coefficients Divisors forestimate of σ $verage c,art /ange c,artn A - / ) / 2 / -

2 1.** J 3.2%( 1.12*3 1. 23 J 2.!(4 1.%+34 .(2+ J 2.2*2 2. !+! .!(( J 2.114 2.32%% .4*3 J 2. 4 2.!34( .41+ . (% 1.+24 2.( 4* .3(3 .13% 1.*%4 2.*4(+ .33( .1*4 1.*1% 2.+(

1 .3 * .223 1.((( 3. (*11 .2*! .2!% 1.(44 3.1(312 .2%% .2*3 1.(1( 3.2!*13 .24+ .3 ( 1.%+3 3.33%

14 .23! .32* 1.%(2 3.4 (1! .223 .34( 1.%!3 3.4(21% .212 .3%3 1.%3( 3.!321( .2 3 .3(* 1.%22 3.!**1* .1+4 .3+1 1.% * 3.%41+ .1*( .4 3 1.!+( 3.%*+2 .1* .41! 1.!*! 3.(3!

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A$$endiF /Coeffi4ients for % ri :les 4h rts ; & s<

Sub grou&si'e C,art for $verages 89

C,art for Standard Deviations 8 s 9

:actors forcontrol limits

Divisors forestimates of

S.D.

:actors for Control limits

N A ) C 2 B ) B 2

2 2.%!+ .(+(+ J 3.2%(

3 1.+!4 .**%2 J 2.!%*4 1.%2* .+213 J 2.2%%! 1.42( .+4 J 2. *+% 1.2*( .+!1! . 3 1.+(( 1.1*2 .+!+4 .11* 1.**2* 1. ++ .+%! .1*! 1.*1!+ 1. 32 .+%+3 .23+ 1.(%1

1 .+(! .+(2( .2*4 1.(1%11 .+2( .+(!4 .321 1.%(+12 .**% .+((% .3!4 1.%4%13 .*! .+(+4 .3*2 1.%1*14 .*1( .+*1 .4 % 1.!+41! .(*+ .+*23 .42* 1.!(21% .(%3 .+*3! .44* 1.!!21( (3+ +*4! 4%% 1 !34

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A$$endiF ECoeffi4ients for % ri :les 4h rts ;Medi n<

Sub grou&si'e C,art for0edians H Control c,arts for /anges ; R 9

:actors forcontrol limits

Divisors forestimate of

S.D.

:actors for control limits

n A - / - / ) / 2

2 1.** 1.12* J 3.2%(3 1.1*( 1.%+3 J 2.!(44 .(+% 2. !+ J 2.2*2! .%+1 2.32% J 2.114% .!4* 2.!34 J 2. 4( .!4* 2.( 4 . (% 1.+24* .433 2.*4( .13% 1.*%4+ .412 2.+( 1.1*4 1.*1%

1 .3%2 3. (* .223 1.(((

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A$$endiF 1Coeffi4ients for % ri :les 4h rts ;Indi%idu ls<

Sub grou&si'e C,art forIndividuals H Control c,arts for /anges ; R 9

:actors forcontrol limits

Divisors forestimate of

S.D.

:actors for control limits

n E - / - / ) / 2

2 2.%% 1.12* J 3.2%(3 1.((2 1.%+3 J 2.!(4

4 1.4!( 2. !+ J 2.2*2! 1.2+ 2.32% J 2.114% 1.1*4 2.!34 J 2. 4( 1.1 + 2.( 4 . (% 1.+24* 1. !4 2.*4( .13% 1.*%4+ 1. 1 2.+( 1.1*4 1.*1%

1 .+(! 3. (* .223 1.(((

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A$$endiF G@ t @ Test ;Criti4 l % lue of @ t @<

Degrees offreedom

nJ1

Confidence Aevel 8M9+ +! ++ ne sided

+ +! ++ T o sided! 1.4* 2. 2 2.!( 3.3% 4. 3( 1.42 1.+ 2.3% 3. 3.!+ 1.3* 1.*3 2.2% 2.*2 3.2!

11 1.3% 1.* 2.2 2.(2 3.111! 1.34 1.(! 2.13 2.% 2.+!1+ 1.33 1.(3 2. + 2.!4 2.*%23 1.31 1.(1 2. ( 2.! 2.*12( 1.31 1.( 2. ! 2.4( 2.((

2+ or more 1.31 1.%4 1.+% 2.33 2.!*

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A$$endiF I@ 1 TestCritical values of 5:6 at +! M Confidence Aevel

Degrees offreedomDenominator

Degrees of freedom numerator

! ( + 12 1! 2 24 3

! !. ! 4.** 4.(( 4.%* 4.%2 4.!% 4.!3 4.!( 3.+( 3.(+ 3.%* 3.!( 3.!1 3.44 3.41 3.3*+ 3.4* 3.2+ 3.1* 3. ( 3. 1 2.+4 2.+ 2.*%12 3.11 2.+1 2.* 2.%+ 2.%2 2.!4 2.!1 2.4(1! 2.+ 2.(1 2.!+ 2.4* 2.4 2.33 2.2+ 2.2!2 2.(1 2.!1 2.3+ 2.2* 2.2 2.12 2. * 2. 424 2.%2 2.42 2.3 2.1* 2.11 2. 3 1.+* 1.+43 2.!3 2.33 2.21 2. + 2. 1 1.+3 1.*+ 1.*4

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A$$endiF V rious $ro: :ilit" /istri:utions

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A$$endiF 9Gloss r" of St tisti4 l Ter#s

T,is document contains t,e basic terms and t,eir definitions related to statistics.

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A44e$t :le Pro4ess Le%el E In acce&tance control c,artingG t,e &rocesslevel fart,est from standard t,at still )ields acce&table &roduct 7ualit) 1 81J α9&ercent of time.

A44e$t :le !u lit" Le%el ;A!L 9 E $ nominal value e &ressed in &ercentdefective or defects &er ,undred units ,ic, ever is a&&licableG for a given grou&of defects or for a single defect.

A44e$t n4e Control J T,e number of lots included in com&utation of t,e &rocessaverage.

A44e$t n4e Line E T,e acce&tance line is a se7uential sam&ling &lan

A44e$t n4e Nu#:er E T,e limiting acce&table value of cG or &n for a sam&le nGt,at isG t,e number of defects or defectives allo ed in a sam&le of si'e n.

A44ur 4" E State of &rocess de&ending u&on ,o close its average is to t,edesired target. /

$ &rocess is said to be $ccurate if t,e value of central tendenc) or mean of itsout&ut is close to t,e targeted value.

A4tion Li#it J Aine on a control c,art be)ond ,ic, t,e &robabilit) of finding anobservation is suc, t,at it indicates t,at a c,ange ,as occurred to t,e &rocess t,at action s,ould be ta en to investigate andQor correct for t,e c,ange.

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C $ :ilit" E M 4hine8 Or Pro4ess E T,e natural tolerance of a mac,ine or&rocess arbitraril) defined to include a&&ro imatel) ++.(3M of all &o&ulation

values.

C $ :le J $ &rocess ,ic, is in statistical control for ,ic, t,e combination ofdegree of random variation t,e abilit) of t,e control &rocedure to detect c,angeis consistent it, t,e re7uirements of t,e s&ecification.

C use & Effe4t /i +r # J $ sim&le tool for individual or grou& &roblemJsolvingt,at uses a gra&,ic descri&tion of t,e various &rocess elements to anal)'e&otential sources of &rocess variation. $lso called fis,bone diagram 8after itsa&&earance9 or Is,i a a diagram 8after its develo&er9.

Centr l Tenden4" E. T,e clustering of a &o&ulation about some &referred value

Centre Line E $ line on a control c,art at t,e value of t,e&rocess mean.

Ch r 4teristi4 E $ distinguis,ing feature of a &rocess or its out&ut on ,ic,variables or attributes data can be collected.

Chi S u re E :or significance of a difference bet een variabilit)6sJn small orlarge σ6 no n.

C i S $li l )& f i &li &l Si l G

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Continuous S #$lin+ Pl n E T)&es of continuous sam&ling &lans SingleGdoubleG CSPJ1 it, sto&Jins&ection ruleG CSP se7uential res&ectivel)

Control E T,e abilit) or need to observeQmonitor a &rocessG record t,e dataobservedG inter&ret t,e data recorded adjust t,e &rocess onl) ,en anadjustment can be justified.

Control Ch rt E $ gra&,ic re&resentation of a c,aracteristic of a &rocessGs,o ing &lotted values of some statistic gat,ered from t,at c,aracteristicG acentral line one or t o control limits. It minimi'es t,e net economic loss fromT)&e I T)&e II errors. It ,as t o basic usesL as a judgment to determine if a&rocess ,as been o&erating in statistical controlG as an o&eration in aid inmaintaining statistical control.

Control Li#its E Aimits on control c,arts calculated on t,e basis of datacollected under controlled conditions. #alues outside t,ese limits indicate t,e&resence of an assignable cause need for corrective action.

Count :le / t E $ form of discrete data ,ere occurrences or events can onl)be counted.

Custo#er s Ris9 J T,e ris of acce&tance ,ic, t,e customer ta es it, t,ea&&lication of a sam&ling &lanG is denoted b) 5 β6

C P E $ &rocess ca&abilit) inde based on t,e ratio of t,e s&read of a fre7uenc)

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/ete4tion J T,e act of discovering.

/e%i tion E Difference bet een desired actual values. In SPC it is used fordifference bet een individual values 8K9 t,eir mean. It is denoted b) 8 9

/is$ersion J T,e s&read or scatter about a central tendenc).

/istri:ution E $ a) of describing t,e out&ut of a commonJ cause s)stem ofvariationG in ,ic, individual values are not &redictable but in ,ic, outcomes asa grou& form a &attern t,at can be described in terms of its locationG s&read s,a&e. Aocation is commonl) e &ressed b) t,e mean or averageG or b) t,emedian s&read is e &ressed in terms of t,e standard deviation or t,e range of asam&le s,a&e involves man) c,aracteristics suc, as s)mmetr) &ea ednessGbut t,ese are often summari'ed b) using t,e name of a common distribution suc,as t,e normalG binomialG or Poisson.

1lo? Ch rt E $ c,art de&icting various ste&s in a &rocess it, t,e ,el& of

standard s)mbols.1re uen4" J :re7uenc) is count of occurrence of an) event. T,at is ,o often anevent occurs in numbers.

1re uen4" /istri:ution J $ table or gra&,G ,ic, dis&la)s ,o fre7uentl) somevaluesG occurs b) com&arison it, ot,ers. Common distributions include normalG

6 t i J 7 ) f d t & d t t il

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6urtosis J :re7uenc) curves of some data are &ea ed near t,e center ,ilet,ose of ot,er data ma) be flat near t,e center. $ fre7uenc) curve ,ic, is&ea ed near t,e center is called Lepto+urtic LG and a fre7uenc) curve t,at is flat

near t,e center is called laty+urtic . T,e normal curve is said to be Meso+urtic.

LAL J Ao er action limit or line

Li#itin+ V lues 1or The Pro4ess A%er +e E T,e u&&er lo er control limitsfor t,e average 7ualit) com&uted from t,e standard 7ualit) level.

Lo4 tion J $ general conce&t for t,e t)&ical values or central tendenc) of adistribution.

Lot K $ &o&ulation of units of an item formed for &ur&oses of ins&ection orcontrol.

Lo?er Control Li#it ;LCL 9 E Control limit on t,e lo er side of t,e central line ina control c,art. $ value belo t,is line indicates &resence of an assignable cause

need for corrective action.Lo?er S$e4ifi4 tion Li#it ;LSL 9 E T,e lo er limit of a tolerance or engineerings&ecifications.

L7L J Ao er arning limit or line

Mid R + ET f t di

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Mid R n+e ET,e average of e treme readings.

Mode E Still anot,er measure of central tendenc) of a set values ,ic, is t,e

value found most fre7uentl) in t,e set.

Mo%in+ Me n J $ mean value calculated from a series of individual values b)moving t,e sam&le for calculations of t,e mean t,roug, t,e series in ste&s of oneindividual value it,out c,anging t,e sam&le si'e.

Mo%in+ R n+e K ;MR< Difference bet een t o consecutive individual valuesused as range ,en number of items in a grou& is one.

Mu ; < J T,e -ree letter used as t,e s)mbol to re&resent t,e true mean of a&o&ulation as o&&osed to t,e various estimates of t,is value ,ic, measurement

calculation made &ossible.

Non Confor#in+ Units E nits ,ic, do not conform to a s&ecification or ot,erins&ection standards sometimes called discre&ant or defective units. P n&

control c,arts are used to anal)'e s)stems &roducing nonJ conforming units.

Non Confor#ities S$e4ifi4 tion E S&ecific occurrences of a condition ,ic,does not conform to s&ecifications or ot,er ins&ection standards sometimescalled discre&ancies or defects. $n individual nonJconforming unit can ,ave t,e&otential for more t,an oneJnon conformities. c u control c,arts are used toanal)'e s)stems &roducing nonJconformities.

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Pro: :ilit" Of Sur%i% l ;EF$onenti l< JT,e reliabilit) of a com&onent ore7ui&mentG ,ic, follo s t,e e &onential failureG &attern.

Pro:le# Sol%in+ J T,e &rocess of moving from s)m&toms to causes 8s&ecial orcommon9 to actions t,at im&rove &erformance. $mong t,e tec,ni7ues t,at canbe used are Pareto c,artG cause effect diagramsG statistical &rocess controltec,ni7ues.

Pro4ess E. $ &rocess is t,e transformation of set of in&utsG ,ic, can includematerialsG actionsG met,ods and o&erationsG into desired out&utsG in t,e form of&roductsG informationG services or generall) results.

Pro4ess A%er +e J T,e location of t,e distribution of measured values ofa &articular &rocess c,aracteristic usuall) designated as an overall averageG KJ

bar.Pro4ess C $ :ilit" J $ measure of t,e ca&abilit) of a &rocess ac,ieved b)assessing t,e statistical state of control of t,e &rocess com&aring t,e amountof random variation &resent it, t,e tolerance allo ed b) t,e s&ecification.

Pro4ess C $ :ilit" IndeF J $n inde of ca&abilit) 8 See C& and C& 9

N N% )P&( = A

!u lit" Ch r 4teristi4 J $ &ro&ert) of a unitG &artG &iece affecting &erformance

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!u lit Ch r 4teristi4 J $ &ro&ert) of a unitG &artG &iece affecting &erformanceor customer satisfactionG suc, as a dimensionG eig,tG or viscosit)G ,ic, ,el&sto differentiate among units of a given sam&le or &o&ulation.

!u rtile /e%i tion J neJ,alf of t,e difference bet een t,e first t,ird7uartile mar s.

!uin4unF >7ui&ment designed to demonstrate t,e distribution &attern ofvariation.

R ndo# C uses J T,e contributions to variation ,ic, are random in t,eirbe,avior.

R ndo#ness J $ state of disorder in ,ic, is not &ossible to &redict anindividual result.

R n+e E Difference bet een t,e largest t,e smallest value in a grou& used as

a measure of dis&ersion or s&read of variation.

R tion l Su:+rou$ J $ subgrou& gat,ered in suc, a manner as to givema imum c,ance for t,e measurements in eac, subgrou& to be ali e t,ema imum c,ance for t,e subgrou&s to differ one from t,e ot,er. T,is subJgrou&ing sc,eme assumes a desire to determine ,et,er or not a &rocess6svariation a&&ears to come from a constant s)stem of c,ance causes.

&o&ulation or universe from ic it is dra n Sam&les s ould be dra n in a

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&o&ulation or universe from ,ic, it is dra n. Sam&les s,ould be dra n in arandom manner.

Sh $e J $ general conce&t for t,e overall &attern formed b) a distribution ofvalues.

SIGMA ; σ < J T,e -ree letter used in SPC to signif) t,e standard deviation of a&o&ulation.

S$e4i l C use J $ source of variation t,at is intermittentG un&redictableGunstable sometimes called an assignable cause. $ &oint be)ond control limits ora run or ot,er nonJ random &attern of &oints it,in t,e control limit signals it.

S #$le SiDe J T,e number of individual results included in a sam&leG or t,e si'eof t,e sam&le ta en. Sam&le si'e is re&resented b) t,e s)mbol 5n6.

S4 tter /i +r # J $ tool to stud) t,e correlation bet een t o variable&arameters. T,e &attern of &oints on a gra&, indicate &resence or absenceG

&ositive or negative strong or ea relations,i&.

S$or di4 Pro:le# J $ &roblem ,en t,e out&ut of a &rocess does notconform to s&ecification re7uirements even ,en t,e &rocess is no n to beca&able. Suc, a &roblem is caused b) a s&ecial or assignable cause.

S$e4ifi4 tion J T,e engineering re7uirement for judging acce&tabilit) of a

Su: Grou$ J ne or more events or measurements used to anal)se t e

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Su: Grou$ J ne or more events or measurements used to anal)se t,e&erformance of a &rocess. /ational subgrou&s are usuall) c,osen so t,at t,evariation re&resented it,in eac, subgrou& is as small as feasible for t,e &rocess

8re&resenting t,e variation from common causes9G so t,at an) c,anges in t,e&rocess &erformance 8i.e.G s&ecial causes9 ill a&&ear as differences bet eensubgrou&s. /ational subgrou&s are t)&icall) made u& of consecutive &iecesGalt,oug, random sam&les are sometimes used.

Str tifi4 tion J Se&arating data &re&aring se&arate ,istograms ,en a,istogram indicates t,at t,e data being studied relates to a mi ed lot. T,e tool,el&s locate reasons for abnormal distribution &atterns.

St nd rd /e%i tion J $ measure of t,e s&read or scatter of a &o&ulationaround its central tendenc). /e&resented b) t,e s)mbol 5 σ6. >stimates of t,estandard deviation are re&resented b) various s)mbols suc, as σnGσ8nJ19 s.

She?h rt Ch rt J T,e control c,arts for attributes variables first &ro&osed b)S,e ,art. T,ese include mean rangeG n&G &G cG u c,arts.

St :le one J T,e central 'one bet een t,e arning limits on a controlc,art it,in ,ic, t,e results are e &ected to fall.

S!C J Statistical 7ualit) control E similar to SPC but it, an em&,asis on &roduct7ualit) less em&,asis on t,e need to control &rocess.

S #$le St nd rd /e%i tion T e sam&le standard deviation in acce&tance

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S #$le St nd rd /e%i tion T,e sam&le standard deviation in acce&tancesam&ling b) variables also an alternative s)mbol for standard deviationgenerall).

Slo$e Of Re+ression Line J T,e slo&e of regression line t,e tangent of t,eangle formed b) t,e regression line KJa is.

Su# T,e o&erator to sumO or to sum ofO.

T ll" Ch rt J $ sim&le tool offered to t,ose ,o are called u&on to record eventsas t,e) occur or to e tract fre7uencies from e isting lists of data.

T r+et J T,e objectives to be ac,ieved against ,ic, &erformances ill beassessedG or t,e mid &oint of a s&ecification.

Toler n4e J T,e difference bet een t,e lo est t,e ,ig,est value stated in t,es&ecification.

Trend $ series of results ,ic, s,o an u& ard or do n ard tendenc).T"$e I Error J /ejecting an assum&tion t,at is true e.g.G ta ing actionsa&&ro&riate for a s&ecial cause ,en in fact t,e &rocess ,as not c,anged overcontrol.

Tot l V ri tion J T,at &rocess variation due to bot, common s&ecial causesG

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A$$endiF L

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J2.% '0''2**(--(=> '0''23-=(2**) '0''2),*3-332 '0''2-*,->--3 '0''2(23)2(>- '0''2'-2*)'3= '0''),'='=*(2 '0'')=,-*'=2' '0'')*>((323' '0'')3=-*2>>3

J2.( '0'')2*='-)'3 '0''))*2-(',= '0'')-*2(2=*2 '0'')(**=*,)) '0'')'=-'()22 '0''-,=,>(>3, '0''->,'(-23( '0''->'->=-(( '0''-=(>'')), '0''-*)32*(2,

J2.* '0''-333(,'*2 '0''-2==()*(* '0''-2'(-2223 '0''-)-=2*-,2 '0''--33=2'(3 '0''-(>*'-33> '0''-((>-*,>' '0''-'3-2-2)2 '0''(,>>22(=2 '0''(,-*-=33-

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UTOMOTIVE SECTOR

J2.+ '0''(>*3>>'(2 '0''(>'=-((') '0''(=3'--23* '0''(*,2>==,> '0''(*2((-,2> '0''(3>>,)>(3 '0''(3)>-*),) '0''(2>,'*=*2 '0''(22()(',2 '0''(),2,3*)*

J3. '0''()2,,*=-- '0''()'*)'=*= '0''(-*),2-*2 '0''(--->)=>= '0''((>-,3,>3 '0''((22-=3>) '0''(('*=3)>- '0''('=')*-33 '0''(')3'=(2= '0''('''>3'=3

J3.1 '0''',*=*=(-2 '0''',)33'22* '0''','2)--*2 '0'''>=2',>*- '0'''>22>'3,- '0'''>(*2(>*> '0'''=>>,((** '0'''=*--*'-) '0'''=)*22')* '0'''=((2->*'

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J3.3 '0'''2>)2>-32 '0'''2**3)=*' '0'''23'(22)) '0'''2)2->*)3 '0'''2(>,2==' '0'''2'2((->= '0''')>,=**=2 '0''')=3>,2*' '0''')*-2>-() '0''')2,3(3),

J3.4 '0'''))*,>'>- '0''')-2>*3-2 '0''')()(33>' '0''')'(>2''- '0'''-,','3=* '0'''->')2(-- '0'''-='()2,( '0'''-*'-=3*= '0'''-3'=3-*3 '0'''-2(333)'

J3.! '0'''-)-*=))= '0'''--2',*,( '0'''-(3>(*-2 '0'''-'=>-(,3 '0'''-''('2,( '0'''(,-*3*-3 '0'''(>32*=)3 '0'''(=>3-,>3 '0'''(=(>)3*) '0'''(*3)=*>(

J3.% '0'''(3,(23=( '0'''(3)()2,) '0'''(2=))=-2 '0'''(2(=23** '0'''()*)3)), '0'''()((3)>3 '0'''(-*(2'*3 '0'''(-()'=*' '0'''((**2>*> '0'''((-(3>'>

J3.( '0'''('=>)'(3 '0'''(')*3,2' '0'''',,*2'32 '0'''',3=*>2( '0'''',-')>'2 '0''''>>2223, '0''''>2,>))> '0''''>(*2,>, '0''''=>2),=( '0''''=3)2>*'

J3.* '0''''=-)=-2) '0''''*,3'=-) '0''''**=2,(- '0''''*2',2)= '0''''*(3),)* '0''''3,'>'3, '0''''3*=(2*= '0''''322)>)) '0''''3--2>)> '0''''3'(2(==

J3.+ '0''''2>((33- '0''''2*(**=* '0''''22-,-=- '0''''2-2,'=' '0''''2'=3>() '0''''),',-2> '0'''')=2,()) '0'''')3,3-)2 '0'''')22=)-2 '0''''))'3(>2

J4. '0'''')(*>*') '0'''')')=)== '0''''-,(()'> '0''''-=,'-'* '0''''-*=)>>= '0''''-3*-(=- '0''''-232>,( '0''''-)3(>== '0''''--3-,=- '0''''-(3>'(,

J4.1 '0''''-'**>=- '0''''(,=,)>3 '0''''(>,32-' '0''''(>(2>22 '0''''(=)=3-= '0''''(**))2* '0''''(3,-(=, '0''''(3-),(- '0''''(23>2)- '0''''(),3*))

J4.2 '0''''())32(' '0''''(-==**) '0''''(----,= '0''''((*,-(> '0''''(((>))= '0''''('*,3*, '0''''('-->-, '0''''',=>')= '0''''',)3((> '0'''''>,),,=

J4.3 '0'''''>32*'- '0'''''>(*>*3 '0'''''=>'=-' '0'''''=2*('- '0'''''=(-,3( '0'''''*>(-'> '0'''''*3'>(* '0'''''*-(=-' '0'''''3,)>*> '0'''''3*=-',J4.4 '0'''''32(*,3 '0'''''3(=-=, '0'''''2,),(= '0'''''2=(3*2 '0'''''23'(>' '0'''''2-,=-2 '0'''''2('(3> '0'''''),(223 '0''''')=)33( '0''''')3*22'

J4.! '0''''')2''>' '0''''')-222' '0''''')',2,' '0'''''-,3-'' '0'''''->(32) '0'''''-*>2,- '0'''''-3*'-( '0'''''-22('* '0'''''-)-=-) '0'''''--(>3'

J4.% '0'''''-((2*2 '0'''''-'(323 '0'''''(,-'=) '0'''''(>)'-, '0'''''(=2),) '0'''''(**(3' '0'''''(3>->' '0'''''(3'=*, '0'''''(2)*'' '0'''''()*=3,

J4.( '0'''''()'-)- '0'''''(-2''2 '0'''''((>'*- '0'''''((-),2 '0'''''('*,>, '0'''''('(>)) '0'''''',*,(* '0'''''',---> '0''''''>==3> '0''''''>)2,=

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J4.+ '0''''''2=,>= '0''''''23*'2 '0''''''2)))3 '0''''''2((=3 '0''''''),(-' '0'''''')=(*) '0'''''')3)'' '0''''''))3-> '0'''''')(>2( '0'''''')'-)=

J!. '0''''''->=(' '0''''''-=-3> '0''''''-3>== '0''''''-23*2 '0''''''-))(3 '0''''''--(-= '0''''''-',,> '0''''''(,,-2 '0''''''(>,'2 '0''''''(=,)2

M!S SPC GL '( Re% '' )' '* '(P +e (*) of (*,

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J!.1 '0''''''(='(- '0''''''(*()* '0''''''(3)'2 '0''''''(23() '0''''''()=*- '0''''''()'2> '0''''''(-)=' '0''''''((=-* '0''''''((((3 '0''''''('3)3

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UTOMOTIVE SECTOR

J!.4 '0'''''''))2' '0''''''')(3> '0'''''''-,>= '0'''''''->-2 '0'''''''-*=' '0'''''''-3-2 '0'''''''-)>* '0'''''''--3* '0'''''''-()- '0'''''''-'(3

J!.! '0'''''''(,'2 '0'''''''(=,, '0'''''''(*,, '0'''''''(*'3 '0'''''''(3(* '0'''''''(2)- '0'''''''()3- '0'''''''(-== '0'''''''(-'* '0'''''''(()>

J!.% '0'''''''('=3 '0'''''''('(2 '0'''''''',3= '0'''''''','2 '0''''''''>3) '0''''''''>'2 '0''''''''=3, '0''''''''=(* '0''''''''*=3 '0''''''''*)=

J!.( '0''''''''*'( '0''''''''3*= '0''''''''3)2 '0''''''''3'2 '0''''''''2=3 '0''''''''22> '0''''''''2-- '0''''''''),> '0'''''''')=3 '0'''''''')3)

J!.* '0''''''''))) '0'''''''')() '0''''''''-,3 '0''''''''-=> '0''''''''-*- '0''''''''-2= '0''''''''-)- '0''''''''-(, '0''''''''-'* '0''''''''(,2

J!.+ '0''''''''(>- '0''''''''(=- '0''''''''(*- '0''''''''(3- '0''''''''(2) '0''''''''()3 '0''''''''(-= '0''''''''((, '0''''''''((- '0''''''''('3

J%. '0''''''''',, '0''''''''',) '0'''''''''>> '0'''''''''>- '0'''''''''== '0'''''''''=) '0'''''''''*> '0'''''''''*2 '0'''''''''*' '0'''''''''3=

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B '0'' '0'( '0'- '0') '0'2 '0'3 '0'* '0'= '0'> '0',

'02 => '03') > )=>> '03'= =>)3)>3 '03(( **3-*3( '03(3 3)2 >> '03( )>>=))) '03-) ---3-*( '03-= ')-2''* '03)(>>(2) *> '03)3>3*23*-*

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UTOMOTIVE SECTOR

. 02,,,,,,,, > 03 ),>,) >>, 03 , >)3)>3 03((, 3 3( 03(3,3)2,>>, 03(,,)>> ))) 03 ), 3 ( 03 , ) 2 03)(>>(2), > 03)3>3 23

.1 '03),>-=>,33) '032)=,3)*2-2 '032==3>2=')' '033(=(*>--,' '0333*='')-*' '033,*(==((3' '03*)33,2=))( '03*=2,2,))2> '03=(2-)=',)- '03=3)232-'('

.2 '03=,-3,*>=(= '03>)(**()2'> '03>='*2)>**= '03,',32'=)-' '03,2>)2>-222 '03,>='*-=)=> '0*'-3*>'3=)2 '0*'*2(,>(2', '0*('-*((>3>, '0*(2',(>(=*=

.3 '0*(=,(()3=2= '0*-(=(,23*3> '0*-33(3=*,*' '0*-,-,,,3232 '0*))'=(*=-,3 '0*)*>)'3>,,2 '0*2'3=*)=2)* '0*22)'>*,>=> '0*2>'-=-),*, '0*3(=)(*==2>

.4 '0*332-(*,*3, '0*3,',*,>333 '0**-=3=-)=(' '0***2'-(2>-( '0*='')(2-'-' '0*=)*22=3>== '0*==-2(>=2() '0*>'>--2>('' '0*>2)>*-,>=( '0*>=,))'3(-3

.! '0*,(2*-2*=)* '0*,2,=2->'3* '0*,>2*>--,(> '0='(,22'3*3' '0='32'(3('*, '0='>>2')22,= '0=(--*')(=3> '0=(3**((,(>2 '0=(,'2-=)*-( '0=--2'2=-2)2

.% '0=-3=2*,)3'* '0=-,'*,(3-2> '0=)-)=((*3,* '0=)3*3-=='(> '0=)>,()=*3(= '0=2-(3),3*)- '0=23)=)(322( '0=2>3=((=3*- '0=3(=2=>2(3, '0=32,'-,=,)=

.( '0=3>')*2-(3- '0=*((2>''*') '0=*2-)=3=*2- '0=*=)'2,>(*= '0==')3''=*-, '0==))=-=-'-= '0==*)=-==,() '0==,)3'(-)>= '0=>-)'2*)('( '0=>3-)*(>-3>

.* '0=>>(22***'* '0=,('-,,=223 '0=,)>,-''*-( '0=,*=)'**3-2 '0=,,323>*',' '0>'-))=3'=,3 '0>'3('33-*3, '0>'=>2,>2-)* '0>('3=')>*(> '0>()-*=',2)'

.+ '0>(3,),,'>-= '0>(>3>>==2,- '0>-(-()*2*)) '0>-)>(22=,== '0>-*),(-)==( '0>->,2)>>==3 '0>)(2=-2'-3, '0>)),=*=3,,, '0>)*23*,2-=) '0>)>,(-,)>3,

1. '0>2()22=2'-2 '0>2)=3-)23-> '0>2*()3=3*(( '0>2>2,2,=,,* '0>3'>)''->=* '0>3)(2',(,(3 '0>332-=*=-2( '0>3=*,')(2)= '0>3,,->>=32( '0>*-(2)),')=

1.1 '0>*2)))>,>2, '0>**3''22))> '0>*>*2)'=-,( '0>='=*(>),(, '0>=->3*=,>3- '0>=2,->'((-3 '0>=*,=332(>( '0>=>,,,23>3, '0>>',,,>))>> '0>>-,=*=2)>(

1.2 '0>>2,)'-*>-> '0>>*>*'2,',- '0>>>=*=2,>,3 '0>,'*3()>)(> '0>,-3(--)=,) '0>,2)3'(*',( '0>,*(*3-3)-( '0>,=,3=*(,(, '0>,,=-=)**2( '0,'(2=2*'33,

1.3 '0,')(,,23'3' '0,'2,'-'(=,( '0,'*3>-2-=2> '0,'>-2'>'(=* '0,',>==-**') '0,((2,(,2>-, '0,()'>2,=,(* '0,(2*3*2,(>( '0,(*-'**-(>= '0,(==)33'=),

1.4 '0,(,-2)->>=2 '0,-'=)'('>33 '0,--(,*(((*( '0,-)*2(222>2 '0,-3'**-3=(, '0,-*2='*,,32 '0,-=>32,-2=( '0,-,-(,'>=-> '0,)'3*))2)*( '0,)(>>=>3(=(

1.! '0,))(,-==(-( '0,)22=>-*)-2 '0,)3=222,'2' '0,)*,,(*(**, '0,)>-(,>'=2( '0,),2-,--,(( '0,2'*-''2,3) '0,2(=,-2)=3( '0,2-,2*3*-,3 '0,22'>-3,*='

1.% '0,23-''=('33 '0,2*)'('==(( '0,2=)>)>*,>( '0,2>22,-*-=3 '0,2,2,=2)'=( '0,3'3->32,'* '0,3(32-=,)=' '0,3-32')2('' '0,3)3-()*==- '0,322>*'3',,1.( '0,332)23*>-- '0,3*)*=',='* '0,3=->)>(3)) '0,3>(>2,'',= '0,3,'='3)-'' '0,3,,2'>>*2) '0,*'=,*(2--2 '0,*(*)*2==)- '0,*-2*-'*,2' '0,*)-=)',*'-

1.* '0,*2'*,=)22, '0,*2>3-(*(>' '0,*3*-'332*) '0,**)=3'>,-3 '0,*=((3,2(3' '0,*=>2)->*=3 '0,*>33=-,,>* '0,*,-3>(33', '0,*,,2*'-*(2 '0,='*-('>*'2

1.+ '0,=(->)3'=(2 '0,=(,))2*('* '0,=-3=(((>*= '0,=)(,**3''3 '0,=)>('--222 '0,=22(-'('-' '0,=3''-(=2>- '0,=33>'>>2>) '0,=*(2>)'3>' '0,=*='2*'-))

2. '0,==-2,,)=,* '0,===>22=3-- '0,=>)'>)=33- '0,=>>-(=,,(, '0,=,)-2,'32- '0,=,>(=>3--2 '0,>')''=,*3( '0,>'==)>,)>= '0,>(-)=-,>=* '0,>(*,((*2)*

2.1 '0,>-()3*2-3> '0,>-3='>>2'* '0,>-,,=')>() '0,>)2(2-3->' '0,>)>--*=2=3 '0,>2---22,)' '0,>2*()=-'2( '0,>2,,**)'** '0,>3)=()-(-2 '0,>3=)=,)(,)

2.2 '0,>*',**'(', '0,>*22=2*3*> '0,>*=,'**((, '0,>=(-*)-(*> '0,>=2323=,=* '0,>===33**3, '0,>>'>,2((>) '0,>>),*-2)=' '0,>>*,*(>>,- '0,>>,>,)=-=3

2.3 '0,>,-=3,(>,2 '0,>,333,2,== '0,>,>-,3>*'' '0,,'',*,2*,2 '0,,')3>(3')= '0,,'*())(-3, '0,,'>*-32>)> '0,,(('3,=('= '0,,()2)*,-2* '0,,(3=3>-->=

2.4 '0,,(>'-2=(() '0,,-'-)=223> '0,,--),=2,', '0,,-23'3>,'- '0,,-*3*)*=)' '0,,->3=(>3)2 '0,,)'3)(2)() '0,,)-22)),(= '0,,)2)'>='3- '0,,)*(->)-=>

2.! '0,,)=,')-'(2 '0,,),*)2-3)) '0,,2()--),** '0,,2-,*>3)'2 '0,,223=)3),* '0,,2*()>-,3' '0,,2=**)*3)> '0,,2,(3'23,- '0,,3'3,,32'* '0,,3-'((=(2-

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2.% '0,,3))>==>-- '0,,32=->3))= '0,,3*')2=22* '0,,3=)'=(==3 '0,,3>32*3>(> '0,,3,=3)*,2) '0,,*',-,-)>* '0,,*-'=),-*' '0,,*)(>>233' '0,,*2-=)3((3

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UTOMOTIVE SECTOR

2.( 0,, 3) , ,3 0,, )3 >, ) 0,, )3>3 ) 0,, >)) ) 0,, , ,> 3 0,, (>(2( 0,, ( ,> 32, 0,, (, ( >, 0,, >(,, ( 0,, ) 23)>3(

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2.+ '0,,>()2((,>* '0,,>(,-=>>,= '0,,>-2,==322 '0,,>)'3(--'- '0,,>)3>>='3- '0,,>2(('*(>3 '0,,>2*(=)*'= '0,,>3(',)-)* '0,,>33>*>,'* '0,,>*'3'2)*2

3. '0,,>*3'')-=> '0,,>*,)*,-)) '0,,>=)*'3=)* '0,,>===(*-() '0,,>>(='2'(3 '0,,>>33=-2(= '0,,>>,)-2*(> '0,,>,-,*)=23 '0,,>,*2,->3) '0,,>,,,(2,-3

3.1 '0,,,')-)->=* '0,,,'*22,332 '0,,,',3*==)* '0,,,(-3,'()> '0,,,(33(,2'> '0,,,(>)3>()- '0,,,-(('>>)2 '0,,,-)==),== '0,,,-*)33,*2 '0,,,->>3=(2'

3.2 '0,,,)(-=,=,- '0,,,))*-*(3' '0,,,)3>,>),' '0,,,)>',>*)) '0,,,2'-->,2> '0,,,2--,()3( '0,,,22->=>'= '0,,,2*--'-)( '0,,,2>','2,( '0,,,2,,''2'*

3.3 '0,,,3(*3(=2* '0,,,3))2*-2' '0,,,32,>33*= '0,,,3*3=()*3 '0,,,3>('3-)' '0,,,3,3>>=() '0,,,*('-))-* '0,,,*-2('32' '0,,,*)=3(=>= '0,,,*3'2>2*(

3.4 '0,,,**)'(,(> '0,,,*=3()2=* '0,,,*>*>22-' '0,,,*,>(3,,> '0,,,=',',2-2 '0,,,=(,*3>=> '0,,,=-,>*3', '0,,,=),=-2)) '0,,,=2,-2=)3 '0,,,=3>222='

3.! '0,,,=*=)-**) '0,,,==3,')', '0,,,=>2(>)=* '0,,,=,-(=>'3 '0,,,=,,>,3', '0,,,>'=)2)=3 '0,,,>(23)-*3 '0,,,>-(2='(3 '0,,,>->(*2)= '0,,,>)2*-)(,

3.% '0,,,>2'>32-, '0,,,>2*>*3'= '0,,,>3-**-=* '0,,,>3>-32)2 '0,,,>*)*2**( '0,,,>*>>2*(3 '0,,,>=)>3,)3 '0,,,>=>*,-2' '0,,,>>))3()- '0,,,>>=>2(,-

3.( '0,,,>,-(*,>3 '0,,,>,*)2'*' '0,,,,'')3,2* '0,,,,'2-)(3, '0,,,,'=,*(,* '0,,,,((3332( '0,,,,(3'(**- '0,,,,(>)3'(( '0,,,,-(3*'-, '0,,,,-2*3(2'

3.* '0,,,,-=*-=3= '0,,,,)'2,-== '0,,,,))-3'>> '0,,,,)3,'3*) '0,,,,)>2*'*2 '0,,,,2',(,2( '0,,,,2)->3)) '0,,,,233*(*= '0,,,,2==3(*- '0,,,,2,>3>-)

3.+ '0,,,,3(>>22> '0,,,,3)>))-2 '0,,,,33='=-> '0,,,,3=3',)' '0,,,,3,-2(>= '0,,,,*','=3- '0,,,,*-3'>*= '0,,,,*2'2=** '0,,,,*33-*=* '0,,,,**,2>(*

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4.+ '0,,,,,,3-'() '0,,,,,,32),* '0,,,,,,3***3 '0,,,,,,3>>-3 '0,,,,,,*'>>' '0,,,,,,*->)= '0,,,,,,*2='' '0,,,,,,**2=- '0,,,,,,*>(3, '0,,,,,,*,=*)

!. '0,,,,,,=(-,' '0,,,,,,=-=2- '0,,,,,,=2(-) '0,,,,,,=32)* '0,,,,,,=**>3 '0,,,,,,==>=) '0,,,,,,=,''- '0,,,,,,>''=* '0,,,,,,>(',* '0,,,,,,>-'**

M!S SPC GL '( Re% '' )' '* '(P +e (** of (*,

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!.1 '0,,,,,,>-,>> '0,,,,,,>)>*2 '0,,,,,,>2*,* '0,,,,,,>32>= '0,,,,,,>*-)> '0,,,,,,>*,3- '0,,,,,,>=*)' '0,,,,,,>>-=2 '0,,,,,,>>>>3 '0,,,,,,>,2*3

!.2 '0,,,,,,,''(= '0,,,,,,,'32' '0,,,,,,,(')* '0,,,,,,,(3'> '0,,,,,,,(,33 '0,,,,,,,-)>' '0,,,,,,,-=>) '0,,,,,,,)(*3 '0,,,,,,,)3-> '0,,,,,,,)>=-

!.3 '0,,,,,,,2(,> '0,,,,,,,23'= '0,,,,,,,2>'( '0,,,,,,,3'=, '0,,,,,,,3)2) '0,,,,,,,33,) '0,,,,,,,3>)' '0,,,,,,,*'32 '0,,,,,,,*-*= '0,,,,,,,*2*,

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UTOMOTIVE SECTOR

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%. '0,,,,,,,,,'( '0,,,,,,,,,'= '0,,,,,,,,,(- '0,,,,,,,,,(> '0,,,,,,,,,-) '0,,,,,,,,,-= '0,,,,,,,,,)- '0,,,,,,,,,)* '0,,,,,,,,,2' '0,,,,,,,,,2)

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UTOMOTIVE SECTOR

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